We briefly review the main stylized facts observed in financial markets and show how a multifractal process naturally captures those effects. In particular we generalize the construction of the multifractal random walk (MRW) due to Bacry, Delour and Muzy to take into account the asymmetric character of the financial returns. We show how one can include in this class of models the observed correlation between past returns and future volatilities, in such a way that the scale invariance properties of the MRW are preserved. Explicit scaling exponents are computes and are shown to behave differently for even and odd moments. We illustrate the usefulness of this "skewed" MRW by computing the resulting shape of the volatility smiles generated by such a process. A large variety of smile surfaces can be reproduced.