The Altai is the highest mountain range in Siberia, located at the very south, where Russia, Kazakhstan, China and Mongolia all come together. The range extends some 300 km west to east and south to north; the highest summit is Mt. Belukha (4506 m, 14783 ft) at the Kazakhstan border.
The area is quite populated by Siberian standards and road network is wide enough to allow access to many rivers without need to hike in. At the same time, coming off the roads you easily get into almost uninhabited places where it's unlikely to see anyone for the whole trip.
The locals are of Mongolian type, doing mainly stock-breeding and little agriculture. They speak a dialect of Turkic but Russian is no problem here too. The major drawback, with roots coming back to soviet times, is that many people are addicted to drink and it's easy to get a conflict with drunks around local settlements. Those who are not addicted to alcohol are really nice and welcoming people.
Type of Paddling
Typically self-supported multi-day runs of any level of difficulty. Most rivers are quite short and require no more than 2-4 days on the water, so it's a good idea to have a car to shuttle between the rivers. The car would be also a restock center, because local food supply is limited. Keep in mind that most rivers do not have road alongside and emergency escape may be difficult.
Season & Climate
July-August. September is still possible, but may be quite cold and wet. Late May and June is also possible but again it's rainy and it's high water season, because most rivers are snow and rain-fed (rivers around Belukha have quite significant glacier drainage). Climate is warm in the summer, but weather may be unstable and unpredictable. Depending on the year you can get very dry summer (and thus low water on most rivers) or very rainy (and high water with sudden floods). You never know what to expect unless you're there, because weather in the mountains is a thing in itself and forecasts do not work. Surprisingly, Altai is almost mosquitoes-free area. Rivers are mainly at moderate altitude of 500-1200 m, only those around Belukha starts at 1500-1700 m.
Getting There & Away
The biggest city in Siberia (and third biggest in Russia), Novosibirsk is located some 600 km north to the Altai Mountains and has numerous flights to virtually everywhere in Russia and many destinations in Europe. Another city, Barnaul, is 300 km closer but has limited number of flights mainly to Moscow. Both have numerous trains connections, although it's a long journey (reckon on 50-60 hrs from Moscow) and probably not a preferable choice.
Local access is done by a car via "Chuya Road", the main route to Mongolia, going from Novosibirsk and Barnaul, via Biysk, Gorno-Altaisk, Onguday, Chibit, Aktash, Kosh-Agach and Tashanta to the Mongolian border. The road is in quite good condition and has numerous local forks, allowing relatively easy access to put-in and take-out for many rivers. These local forks are often rough and in most cases a 4x4 vehicle is necessary.
Most of the area is free of any paperwork, except for small corner next to Chinese and Kazakhstan border, a borderland permit is required to go there. There are just a couple of rivers there, so extra fuss is unlikely worth doing.
Some valleys belong to the Altai Nature Reserve, and in theory you need appropriate permit to enter. The regulations are not strict though; you can easily get this in nearest office or directly on the spot for a small fee (about $1-2 per person per day).