Consider an elliptic curve $E$ defined over a number field $K$. The set $E(K)$ of $K$-points is a finitely generated abelian group whose rank, called the rank of $E$, is an important invariant. It is an open and difficult problem to determine which ranks occur for elliptic curves over a fixed number field $K$. We will discuss recent work which shows that there are infinitely many elliptic curves over $K$ of rank $r$ for each integer $0\leq r \leq 4$. We will make use of explicit families of curves and a 2-descent.