In this paper we determine possible decompositions of Euler polynomials E_k(x), i.e. possible ways of writing Euler polynomials as a functional composition of polynomials of lower degree. Using this result together with the well-known criterion of Bilu and Tichy, we prove that the Diophantine equation -1^k +2 ^k - ... + (-1)^x x^k=g(y), with g∈Q[X] of degree at least 2 and k≥ 7, has only finitely many integers solutions x, y unless polynomial g can be decomposed in ways that we list explicitly.