  A Banach space is {\it polynomially Schur} if sequential convergence against
analytic polynomials implies norm convergence. Carne, Cole and Gamelin show
that a space has this property and the Dunford-Pettis property if and only if
it is Schur. Herein is defined a reasonable generalization of the
Dunford--Pettis property using polynomials of a fixed homogeneity. It is shown,
for example, that a Banach space will has the $P_N$ Dunford--Pettis property if
and only if every weakly compact $N-$homogeneous polynomial (in the sense of
Ryan) on the space is completely continuous. A certain geometric condition,
involving estimates on spreading models and implied by nontrivial type, is
shown to be sufficient to imply that a space is polynomially Schur.
