  We prove mean convergence, as $N\to\infty$, for the multiple ergodic averages
$\frac{1}{N}\sum_{n=1}^N f_1(T_1^{p_1(n)}x)... f_\ell(T_\ell^{p_\ell(n)}x)$,
where $p_1,...,p_\ell$ are integer polynomials with distinct degrees, and
$T_1,...,T_\ell$ are commuting, invertible measure preserving transformations,
acting on the same probability space. This establishes several cases of a
conjecture of Bergelson and Leibman, that complement the case of linear
polynomials, recently established by Tao. Furthermore, we show that, unlike the
case of linear polynomials, for polynomials of distinct degrees, the
corresponding characteristic factors are mixtures of inverse limits of
nilsystems. We use this particular structure, together with some
equidistribution results on nilmanifolds, to give an application to multiple
recurrence and a corresponding one to combinatorics.
