In this talk we introduce a new class of special functions, called Bernstein-gamma functions. This simple class extends the classical Gamma function. We derive the Stirling asymptotic and describe the main analytic properties of the class of Bernstein-gamma functions. We then use these functions to evaluate explicitly the Mellin transform of the so-called exponential functionals of Levy processes which in turn allows a thorough study of these random variables. We also discuss the link between the Bernstein-gamma functions and many quantities related to one-dimensional self-similar Markov processes. The latter suggests that Bernstein-gamma functions can play the same role the Gamma function plays in the understanding of many continuous path stochastic processes. This is joint work with Pierre Patie (Cornel, US).