【Abstract】In this talk, we begin with the Ito & McKean's construction of Skew B.M. (via the Brownian excursions), and introduce a class of the so-called Skew diffusion processes. Specifically, our considerations are limited on the Skew O-U processes and the Skew Feller-branching processes (the latter are also called Skew CIR processes). For those two processes we first give the explicit expressions on the transition densities, in term of Special Functions. Next we study the hitting times of the processes up (or down) crossing some given levels, and we obtained the Laplace Transforms expressions of those random stopping times. These results are fundamental for the quantitative analysis of the processes. On the other hand, some observations from the FX market data show that, the special structures of Skew O-U processes can capture the important "sticky" phenomena, which frequently appeared in the market while the FX prices go up (down) to some specific levels. Whereas the usual Geometric BM or Geometric O-U processes fails to do. So with the nice tractable characters the Skew O-U procesess can be significantly introduced to model some FX and other assets price dynamics, alternatively we can proceed to the derivative securities pricing with those models.