The method of fundamental solutions (MFS for short) is a meshfree numerical solver for linear and homogeneous partial differential equations and has been applied to several problems such as Laplace equation, Helmholtz equation, or biharmonic equation. The robust feature of the MFS is that under some ``nice'' conditions the approximation error decays exponentially for the number of approximation points. However, it is challenging to describe ``nice'' conditions, and the MFS is applied to several problems in engineering without mathematical proofs. In this talk, we talk about the mathematical theory on the MFS applied to Helmholtz-type equations as a milestone for constructing a unified mathematical theory on the MFS.