If A is an abelian variety over a p-adic field K then A has good reduction if and only if the p-adic Tate module of A is a crystalline representation of the absolute Galois group of K. As there are examples of curves over K with bad reduction whose Jacobian has good reduction, the Galois action on the p-adic etale cohomology of the curve does not determine its reduction. We will discuss these issues and point to a p-adic criterion of good reduction for curves.