  We introduce a method for solving Calder\'on type inverse problems for
semilinear equations with power type nonlinearities. The method is based on
higher order linearizations, and it allows one to solve inverse problems for
certain nonlinear equations in cases where the solution for a corresponding
linear equation is not known. Assuming the knowledge of a nonlinear
Dirichlet-to-Neumann map, we determine both a potential and a conformal
manifold simultaneously in dimension $2$, and a potential on transversally
anisotropic manifolds in dimensions $n \geq 3$. In the Euclidean case, we show
that one can solve the Calder\'on problem for certain semilinear equations in a
surprisingly simple way without using complex geometrical optics solutions.
