  This overview article concerns the notion of fractional smoothness of random
variables of the form $g(X_T)$, where $X=(X_t)_{t\in [0,T]}$ is a certain
diffusion process. We review the connection to the real interpolation theory,
give examples and applications of this concept. The applications in stochastic
finance mainly concern the analysis of discrete time hedging errors. We close
the review by indicating some further developments.
