In recent years, ideas about the fundamental difference between comets and asteroids have been significantly revised. Several dozen asteroids of the Main Belt showed signs of cometary activity (comas, tails), so they were named "comets of the main belt". With ground-based instruments, such active asteroids were detected mainly by imaging, and only sufficiently powerful manifestations of the activity of asteroids of the Main Belt were observed. Undoubtedly, the number of asteroids in which activity occurs on a more moderate scale is incomparably greater. New approaches are needed to register weak activity using ground-based instruments. Such an effective approach turned out to be the use of the traditional method of broadband photometry and low-resolution spectroscopy. In the papers [1, 2, 3, 4], the application of this approach made it possible to massively study a weak form of activity of primitive types of asteroids, called sublimation dust activity (SPA). SPA leads to the formation of a temporary, rather sparse dust exosphere of asteroid, which can be detected by broadband photometry. The review provides up-to-date information about active asteroids. The main attention is paid to the manifestations of SPA and light scattering models in the dusty exospheres of asteroids. The mass character of the SPA and possible mechanisms responsible for the SPA are discussed.