The general problem of a two-dimensional water-solid impact is considered. The method of matched asymptotic expansions is used to solve the resulting boundary value problem for small penetration depth. The first-order solution obtained extends the classical Von Kármán and Wagner theories to non-normal impact and initially curved free surfaces. It accounts for nonlinear features such as the creation of a jet at each waterline. The application of the method to the computation of the impact loads exerted by spilling breakers on a marine structure is discussed.