\[30\], \[19\] and non-symmetric stable processes \[20\]-\[22\]. In the present article with the help of M.Kac’s idea \[11\] and the theory of integral equations with the difference kernels \[22\] we investigate a wide class of Levy processes. We note that within the last ten years the Levy processes have got a number of new important applications, particularly to financial problems. We consider separately the examples of Levy processes which are used in financial mathematics.\
Now we shall formulate the main results of the article.\
1. The Levy process $X_{t}$ defines a strongly continuous semigroup $P_{t}$ (see \[23\]). The generator $L$ of the semigroup $P_{t}$ is a pseudo-differential operator. We show that for a broad class of the Levy processes the generator $L$ can be represented in a convolution type form (section 2): $$Lf=\frac{d}{dx}S\frac{d}{dx}f,$$ where the operator $S$ is defined by the relation $$Sf=\frac{1}{2}Af+\int_{-\infty}^{\infty}k(y-x)f(y)dy,$$ Such a representation opens a possibility to use the theory of integral equations with difference kernels \[22\].\
2. We introduce the notion of the truncated generator $L_{\Delta}$, which is important in our theory and coincides with $L$ on the bounded system of segments $\Delta$. We define the quasi-potential $B$ by the relation $-L_{\Delta}Bf=f$ (sections 3 and 4). 