Inverse problems of recovering the metric and nonlinear terms were originated in the work by Kurylev, Lassas, and Uhlmann for the semi-linear wave equation $\square_g u(x) + a(x)u^2(x) = f(x)$ in a manifold without boundary. The idea is to use the multi-fold linearization and the nonlinear interactions of distorted planes waves to produce point-source-like singularities in an observable set. In this talk, I will discuss joint works with Gunther Uhlmann, which consider the recovery of the nonlinearity for a quasilinear wave equation arising in nonlinear acoustic imaging. The main difficulty that we need to handle here is caused by the presence of the boundary.