This is a nicely written survey article on the conjectures in the title of the paper. The conjectures of Serre in question are about the modularity of mod\,$p$, 2-dimensional, continuous, odd, absolutely irreducible representations of the absolute Galois group $G_ Q$ of $ Q$. There is a more refined version which also predicts certain minimal modular invariants from which these Galois representations arise. While the conjectures in their qualitative form are still wide open there has been considerable progress in proving that the qualitative form of the conjecture implies the refined form. It is this implication, which is a consequence of deep work of many mathematicians, that this paper surveys in the main. The paper also has useful exercises that will be of help to someone wishing to learn about this area, and two appendices by K. Buzzard and B. Conrad on mod $ l$ multiplicity one principles and constructions of Galois representations attached to weight 2 newforms.
{For the entire collection see MR 2002d:11003.}

Reviewed by Chandrashekhar Khare