The theory of polarizable Hodge modules was developed by M. Saito at the end of the last century. The theory of twistor structures was then developed by C. Simpson. It is expected that many parts of the Saito theory will have an appropriate analogue in a more general case of mixed twistor structures. In particular, there is the Kashiwara conjecture for push-forwards and vanishing cycles in the case of semisimple holonomic D-modules. This book is devoted to the Kashiwara conjecture for the category of polarized regular twistor D-modules (equivalent to the category of semisimple perverse sheaves). It introduces a notion of regular twistor D-modules and it contains proofs of the corresponding versions of the decomposition and the vanishing cycles theorems. The main strategy used for the proof is similar to that of Saito theory. One of the tools used in the proofs is the Mellin transform for distributions developed by D. Barlet and M. Kashiwara. There is no doubt that the work available now in the revised book presents an important step on the way to the proof of the Kashiwara conjecture.