Let F_q be a finite field, and consider the set P^2(F_q) of all F_q-points in the projective plane. A subset B of P^2(F_q) is called a blocking set if B meets every line defined over F_q. Given an algebraic plane curve C in P^2, when does the set of F_q-rational points on C form a blocking set? We will see that curves of low degree do not give rise to blocking sets. As an example of this principle, we will show that cubic plane curves defined over F_q do not give rise to blocking sets whenever q is at least 5. On the other hand, we will describe explicit constructions of smooth plane curves (of large degree) that do give rise to blocking sets. Finding blocking curves of optimal degree over a given finite field remains open. This is joint work with Dragos Ghioca and Chi Hoi Yip.