One special case of the Birch--Swinnerton-Dyer conjecture is the statement that if E is an elliptic curve over a number field, and the L-function of E does not vanish at s = 1, then E has only finitely many rational points and its Tate-Shafarevich group is finite. This is known to be true for elliptic curves over Q by a theorem of Kolyvagin. Kolyvagin's proof relies on an object called an 'Euler system' -- a system of elements of Galois cohomology groups -- in order to control the Tate-Shafarevich group. It has long been conjectured that Euler systems should exist in other contexts, and these should have similarly rich arithmetical applications; but only a very small number of examples have so far been found. In this talk I'll describe the construction of a new Euler system attached to pairs of elliptic curves -- or more generally pairs of modular forms -- and its arithmetical applications to the Birch--Swinnerton-Dyer conjecture. This is joint work with Antonio Lei and David Loeffler, and it has recently been generalised by Guido Kings, David Loeffler and myself.