This is new to me, so I will impart what little knowledge I have!! - Just a few photos to start;
UV photography is much more specialised, and expensive, than normal light or even Infrared photography, you need four specialist pieces of equipment 1) a UV sensitive camera 2) a Lens that transmits UV light 3) a Filter to block all visible and Infrared light and 4) a strong UV light source.
The camera is normally a Full Spectrum converted Mirrorless or Live View camera, a traditional DSLR is not going to work at all, as you won't see anything at all through the optical viewfinder. The other requirements, like full frame or APS-c format, rear screen or evf etc. apply as for normal or IR photography. Your probably going to be using high ISO settings, so a low noise sensor is going to be an advantage. I use my trusty Full Spectrum Sony A7R, but any of my Full Spectrum converted cameras in the Shop will be suitable.
The lens is probably the most important piece of equipment to get right, the vast majority of lenses are designed not to pass any UV at all, modern glass and coatings block 99% of UV light as it tends to impair the visible light image. There are special lenses that are designed for UV imaging, the most famous 2 are the 105mm UV Nikkor and the 60mm Coastal Optics lenses, both of these are very expensive £3-6000!! There are others but they are all very expensive. But as luck would have it there are several accidental UV passing lenses, initially it was thought that the 63mm EL-Nikkor enlarging lens was suitable and it does pass a reasonable amount of UV, but test by several people in the UV community have shown that the older 80mm f5.6 EL Nikkor is the best of the Nikon enlarging lenses. These lenses however do not have any focusing mechanism, so have to be adapted to camera use with bellows or focusing helicoids.
These same UV enthusiasts have also discovered that some very old pre multi coating era lenses from the 50's-70's can pass UV. The simpler the lens design the better, so less lens elements, single coating, no cemented lens groups. A particular manufacturer in Japan, Kyoei Kuribayashi made a 35mm f/3.5 lens that is particularly suitable, it passes UV strongly from about 320nm upto 400nm where visible Indigo takes over. It is not as good as the specialist UV lenses, but certainly usable and generally better than the enlarging lens option. The only problem is finding one!! Luckily the manufacturer made these lenses for dozens of different Brand Names, but identifying which lenses where made by Kyoei Kuribayashi to the correct design is a minefield to say the least. I have now made life easier as I have sourced a few of these old lenses, as I now know what to look for, and have listed them on the Shop pages along with kits that include various adaptors. Other lenses and filter combinations also work to a lesser extent, they won't be as good and you may well need to use even higher iso's.
The Filter for the lens is equally important, as it needs to pass at least as much UV as the lens can pass and sensor can record, but it must block all visible and especially IR light. The top image above is with a combination of two filters a ZBW1 [Schott UG11 Equivalent] filter that passes UV from about 330nm to 400nm, but unfortunately also passes IR light above 700nm. A secondary BG39 filter is added to block this unwanted IR as it would swamp the UV and the shot would be basically an IR shot. In daylight conditions there is always massively more IR than UV, plus most lenses are not terribly efficient at passing UV, so its essential to block all visible and IR light.
The other 3 photos are with a similar lens, but a different type of UV passing filter, a Baader U filter, which is basically a similar filter to the UV passing filter in the top shot, but it has been especially coated to block all IR. This is much more efficient at passing UV light and so allows much shorter exposures, unfortunately they are only available in 1.25" and 2" sizes for telescopes, plus they are very expensive. A 2", which is actually a 48mm filter size, is about £265, but it is the filter to get as it allows reasonable exposures in sunlight. The cheaper option of a UV passing and IR blocking filter combination, such as a UG11 + BG 39 filter stack lets about half as much UV through, but costs about half as much and the filter glass of the UG11 especially is prone to oxidation, so has to be treated very carefully. The Baader U filter has its own problems, the IR reflecting coatings are very shiny, one side is yellowy the other side is pinkish. How it comes, the yellowy side faces the subject, this causes major flare problems, but if you reverse the glass in its mount so the pinkish side faces the subject the problem is much reduced. The Baader U does not suffer from oxidation problems however, because these very coatings protect the raw glass underneath. Again we will reverse the glass for you in the Baader U filters listed in the Shop.
The light source is very often the Sun, but can also be modified flashguns or specific UV lights. The Sun is a very good source of UV, but as UK photographers know only too well is extremely variable. ALL SOURCES OF UV MUST BE TREATED WITH EXTREME CARE!!! You wouldn't stare into the sun, but you can easily find yourself looking directly into a UV light source without knowing, its UV, you can't see it!! If you do you may well find you can't see anything at all, ever!! Its not good for your skin either, so you have been warned. Don't mess about with UV light sources and don't subject your model's eyes or skin to artificial UV either. Sunglasses are not good enough protection, get proper UV goggles or safety protection glasses.
Oh, and your probably going to need a tripod, you can just about hand hold a Kyoei style 35mm lens at f3.5 with a Baader U filter in strong sunlight, but you will need to be using 3200iso or so.
White Balance is equally important as when shooting false colour IR in jpeg format, again Raw files can be balanced in programs like Capture One in post production.