The main theorem states that a system of r quadratic forms in n variables defined over an arbitrary p-adic field Khas a nontrivial zero over K as long as n > 4r. This bound is optimal. This result answers a question first posed over 50 years ago by Birch and Lewis. The theorem is obtained from a much more general result dealing with systems of quadratic forms defined over a complete discretely valuedfield K with residue field k. Namely, if there is a constant A such that for each r, every system of r quadratic forms defined over k in more than Ar variables has a nontrivial zero over k, then every system of r quadratic forms defined over K in more than 2Ar variables has a nontrivial zero over K. The theorem on systems of quadratic forms defined over a p-adic field follows as a corollary with A=2 by a theorem of Chevalley applied to systems of quadratic forms defined over finite fields.The theoremhas important applications to computing u-invariants of arbitrary function fields over p-adic fields and other complete discretely valued fields.