Summary: A special class of diffusion processes, the generalized hyperbolic diffusion processes, is introduced. As a byproduct we present a technique for the construction of one-dimensional ergodic diffusion processes with a predetermined stationary density. We specifically study the application of this new type of diffusion process to financial data, especially U.S. stock prices. It is seen that in addition to confirming stylized features of the financial market, a key explanation concerning “thick-”tailed log returns is provided.