This thesis is devoted to the study of inverse problems for semilinear elliptic equations on Stein manifolds with Kähler metric. After developing some preliminary techniques, we will show that the Dirichlet-Neumann maps for certain semilinear elliptic equations determine the nonlinearities. We will consider two inverse problems of this kind with distinct geometric conditions imposed. The first one is the inverse problem for nonlinear Schrödinger equations on Kähler manifolds having specific Stein-like properties. The second one is the inverse problem for nonlinear magnetic Schrödinger equations on Riemann surfaces with partial data boundary measurements. In both cases, the nonlinearities involved are assumed to have certain analytic representations and vanishing lower order terms. The key observation is that, by a suitable linearisation procedure, one could transform the nonlinear problems into series of linear problems which have close connections to the techniques we develop.