When I first told my wife that cacti grew wild in Alberta, Canada, she didn't believe me. Then we went out to Drumheller and found some Opuntia polyacantha. In addition to Alberta's three native cacti, there are a number of others that can be grown in gardens here. When we got a house, I built a rockery in my south facing back garden and planted some cacti and succulents in it. I've also had some experience of growing cacti in England, which has a warmer but wetter climate.
This article's going to use some Latin names of plants. You can easily look them up on the Internet. It will mainly deal with two groups of cacti (family Cactaceae): opuntoides (sub-family Opuntoideae), which are basically the prickly pears, chollas and club chollas and globular cacti (an informal sub-group of the sub-family Cactoideae), which are those small round cacti that have pretty flowers (read more about them). However Maihuenia is the only genus of the sub-family Maihuenioidea. These are all desert type cacti, the forest cacti (Pereskia, Schlumbergera etc.) aren't very hardy.
Perhaps it's fairly obvious but to grow cacti outdoors in a cold climate, you really need a well-drained site, like a rockery (although it rather depends on how wet your climate is). You should also have the main slope of your rockery facing south (in the northern hemisphere) so it makes more effective use of the sun's rays (in nature cacti are often found on south facing slopes) The type of soil isn't that important. My rockery uses a mixture of peat, loam and gravel, according to advice for alpine rockeries. If your local soil is alkaline, it would help to make it more acidic (e.g. by adding peat). This site should also be well lit and preferably with a building or something on the north side to shelter the plants from the harsh north wind.
You should leave a lot of space around opuntoids as they tend to grow very fast and also have nasty spines and barbed hairs called glochids. You could plant Sempervivums (hardy leaf succulents often used as place holders on alpine rockeries) around them, which can be removed later. Globular cacti (and probably Maihuenias) need to be watched to make sure they don't get smothered. You can add variety by having other succulents on the rockery: Sempervivum, Sedum, Lewisia etc.
I once met a woman who claimed that she'd grown Myrtilocactus geometrizans (one of the large tree like cacti - not known for its hardiness) outdoors in northern England for a number of years. However you really should stick to hardier cacti and your choice will depend on your climate.
Probably the hardiest cactus is Opuntia fragilis, which has a huge distribution range over much of North America going up to near the artic circle. This is a small prickly pear that forms mats of very fleshy, almost cylindrical joints (pads technically cladodes, these are actually sections of stem, not leaves) each about 3cm long. The name comes from the fact that its joints are very easy to detach, perhaps making it a bad plant for high traffic areas. There are a number of clones and hybrids with various shaped joints and degrees of spinyness. It normally has yellow flowers but some forms have magenta flowers. I've had two forms for a number of years. One comes from Peace River, Alberta, where it's said that all the plants are the same clone and they seldom flower (all opuntoides seem to be self-incompatible so you need at least two clones to produce seeds). I don't know the extent of the genetic research but I've yet to see this flower. I have another form that was sent from Ontario (presumably a form that grows wild there) and does flower. I've recently bought a specimen of the Okanagan Valley (British Columbia) form, which has almost spherical joints and seems to branch more prolifically and grow more upright with fewer spines. I also have a spineless plant that looks like a small form of O. basilaris that prolifically produces yellow flowers. (O. basilaris isn't hardy in Calgary but the plant I'm talking about is and is presumably a form or hybrid or O. fragilis although it's been suggested that it's an O. aurea.)
Alberta's other Opuntia is O. polyacantha. This has pads about 10cm long (rather varied depending on genetics and environment) and more obviously flat than in O. fragilis. It also grows along the ground to form clumps. As the name suggests, it's very spiny. Again the flowers are normally yellow but it has magenta flowered forms and I've heard of very dark flowered plants in Alberta but haven't seen them (I've also heard of orange flowered plants but haven't seen one flowering).
Alberta's only other cactus is Escobaria vivipara (often called Coryphantha vivipara). This is a small globular cactus with spiny tuberculate stems and large magenta flowers. I've never figured out how to grow this from seed (I've successfully raised other species of Escobaria along with many other types of cacti and succulents). It presumably likes cooler conditions but I've never managed to get it right. I planted three seedlings that I bought from a native plant company on my rockery a number of years ago. The one survivor is still very small.
I've also found a number of hardy Opuntia hybrids: 'Claude Arno', 'Pony' and 'Peifky' are rather similar, like smaller O. polyacantha with magenta flowers. 'Namao Rose' looks like a small form of O. basilaris, again with magenta flowers. In fact many Opuntia hybrids have the beautiful O. basilaris in their ancestry for the magenta flowers.
It would be nice to have huge prickly pears, towering columnar cacti or larger Schick Echinopsis hybrids in my garden. Unfortunately all the hardy cacti seem pretty small. Cylindropuntia (North American Chollas) are taller than most. These are similar to Opuntia but with sausage shaped joints and often very nasty spines. I managed to keep C. imbricata in my rockery for several years. C. viridiflora seems to be even hardier and still growing there. The hardiest Cylindropuntia is supposed to be C. davisii but I've never been able to get hold of one (it's supposed to be a natural hybrid of uncertain parentage, there's only one clone and it's self incompatible or sterile so you can't get seeds).
The only other cactus that has lasted long on my rockery has been Pediocactus simpsonii, eventually being finished off by an unusually cold, wet spring. This is another globular cactus, this time with whitish flowers. Pediocacti aren't too difficult to grow from seed and there are several species and seed is available from many populations, giving you room for experimentation. Unfortunately most of them are CITES A listed, meaning that they're officially endangered and special (and costly) permits are required for importing seed.
Calgary is a rather harsh climate, in zone 3 (which means it sometimes goes down to 40 where Fahrenheit and Centigrade are equal) and there are Chinooks causing rapid changes in temperature in the winter. If you live in a slightly more forgiving climate (or find a tougher form or clone than I've managed to get or a species I haven't tried) there are a number of other cacti, which might work for you.
There are a number of Opuntias famed for their hardiness (the classification is a bit confused). O. humifusa (often called O. compressa) has flat joints, which are usually spineless (but with glochids). O. macrorhiza is similar but tends to produce tuberous roots. O. chymochila, O. pheanacantha and O. erinaceae are very spiny plants, similar to O. polyacantha. O. aurea is similar to O. balisaris with clumps of upright joints branching from the base, normally spineless, only it usually has yellow flowers (O. basilaris usually has magenta flowers). O. 'rutilans' and O. 'super rutila' are rather confused and different sources disagree on if they're the same thing and if it's a hybrid or a species (O. debreczyi). They're similar to spineless O. fragilis with red flowers (I have O. 'super rutila' but haven't tried it in my rocker or seen it flower). Opuntias are easy to grow from cuttings (seed is more problematic), allowing you to raise spare plants for experiments.
Some Grusonias (club chollas opuntoids sometimes classified in other genera) may be worth investigating. They're similar to small Cylindopuntias.
Another very hardy globular cactus is Escobaria missouriensis (sometimes called Neobessia missouriensis). This has greenish flowers. Like E. vivipara, I've never figured out how to grow it from seed, but I haven't tried it in my rockery either.
Another genus known for its hardiness is Echinocereus. These are mostly small, cylindrical stemmed cacti with large magenta flowers. The hardiest species is probably E. viridiflora, which has small green flowers. I've tried this in my rockery but it didn't last long. I have seen it growing outside in Edmonton (farther north than Calgary but at a lower altitude and doesn't get Chinooks). The next hardiest species is probably E. triglochidatus. This is one of the larger species forming clumps of cylindrical stems with about 6 ribs and red cup shaped flowers (hence the common name: claret cup cactus). Echinocereus are easy to grow from seed and you can get seeds from different locations for variety and trying to find an extra hardy form.
Mammillaria is the biggest genus of North American globular cacti, they're very varied in appearance and cultivation requirements (read more about them). They hardiest type is probably M. heyderi var. bullingtoniana, which has dirty white flowers.
So far, all these cacti have been North American. For some reason South America doesn't seem to have produced any very hardy cacti (the only cactus thought to occure naturally outside the Americas is an epiphyte - not very hardy). The hardiest South American cacti are probably Austrocactus (rare and difficult to grow from seed), Maihuenia (very strange cacti M. poepigii seems tolerant of winter wet so it may be a good one for England), Rebutia, Oroya and Lobivia (now included in Echinopsis but DNA evidence seems to suggest they're a distinct group). The last four are easy to grow from seed and Rebutia and Lobivia will also grow from cuttings. Gymnocalycium bruchii and G. baldianum are also fairly hardy and easy to grow from seed. There are also some fairly tough South American opuntoides (almost impossible to obtain in Canada and hard to grow from seed): Airampoa (or should it be Tunilla?), Maihueniopsis, Tephrocactus, Austrocylindropuntia and Pterocactus. If you can get these, they're mostly easy to grow from cuttings so you can experiment.
You can sometimes get these plants from your local garden centres. If there's a local Cactus and Succulent society, you may well be able to get plants from there (most cactus collectors have more Opuntia cuttings than they know what to do with). You can also order plants and seeds from specialist sources, which can be found on the internet. You may be able to collect plants (whole or cuttings) from the wild but you should check the applicable laws. If you live in or near a cactus habitat, you be able to get plants that have been rescued due to development of the habitat. There are also laws about moving plants (and less so seeds) across international borders (this is particularly an issue with Pediocactus).
I hope I've inspired you to grow some cacti as hardy perennials, as they will add interest to most gardens.