Some examples of weight 3 motives with Hodge numbers 1,1,1,1

This talk will be an introduction to pure motives and to the generalized Birch and Swinnerton-Dyer conjecture relating the order of vanishing of an L-function at the center of the critical strip to the rank of a Chow group. The purpose of the talk is to give general information about the field. No new results will be presented. In keeping with the spirit of the seminar the examples will be taken from motives coming from threefolds whose cohomology has Hodge numbers $h^{30}=h^{21}=h^{12}=h^{03}=1$.