The review discusses chemical mechanisms of most important reactions that take place in the course of catalytic cracking of alkenes and alkanes over solid acidic catalysts. The main subject of the review is the mechanism of the principal cracking step, fission of C-C bonds in aliphatic hydrocarbons. The cracking mechanism of alkenes via carbenium ion intermediates (with complications caused by oligomerization reactions) is well established. In contrast, the C-C bond fission mechanism in alkanes is still a controversial subject. The review compares merits and difficulties of different mechanisms proposed in the Literature: several carbeniun-ion mechanisms, the carbonium-ion mechanism, the chain-reactions mechanism with participation of both carbonium and carbenium ions, and the oxonium-ion mechanism. Reactions accompanying C-C-bond fission in cracking reactions, hydrogen redistribution, formation of cycloalkanes, Light aromatic compounds, and coke are also briefly discussed.