The method of fundamental solutions (MFS) is a boundary method used for solving elliptic boundary value problems. It is applicable when a fundamental solution of the differential equation is known and shares advantages with the boundary element method (BEM). This paper reviews the development of MFS and related methods over the last three decades, highlighting their applications and extensions to non-trivial problems. The MFS approximates the solution as a linear combination of fundamental solutions, with singularities placed outside the domain. The locations of these singularities are either preassigned or determined along with the coefficients to satisfy boundary conditions. The solution is typically obtained by a least squares fit of boundary data. If the singularities are to be determined, the resulting minimization problem is nonlinear and can be solved with available software. The MFS can be extended using an auxiliary boundary, which helps avoid singularities in boundary integral equations. This approach has been used in various problems, including potential problems, elastostatics, and biharmonic problems. The method of Kupradze uses observation points on an auxiliary boundary to avoid singularities in boundary integral equations. Oliveira introduced the idea of using a simple layer potential on an auxiliary boundary, which has been further explored. The desingularization technique, moving observation points away from the boundary, is also used to solve problems in inviscid flow, biharmonic problems, and spherical shell problems. In the case of Laplace's equation, the solution is expressed as an integral over an auxiliary boundary using a simple layer potential. This method is effective for solving elliptic boundary value problems and has been widely applied in various fields.The method of fundamental solutions (MFS) is a boundary method used for solving elliptic boundary value problems. It is applicable when a fundamental solution of the differential equation is known and shares advantages with the boundary element method (BEM). This paper reviews the development of MFS and related methods over the last three decades, highlighting their applications and extensions to non-trivial problems. The MFS approximates the solution as a linear combination of fundamental solutions, with singularities placed outside the domain. The locations of these singularities are either preassigned or determined along with the coefficients to satisfy boundary conditions. The solution is typically obtained by a least squares fit of boundary data. If the singularities are to be determined, the resulting minimization problem is nonlinear and can be solved with available software. The MFS can be extended using an auxiliary boundary, which helps avoid singularities in boundary integral equations. This approach has been used in various problems, including potential problems, elastostatics, and biharmonic problems. The method of Kupradze uses observation points on an auxiliary boundary to avoid singularities in boundary integral equations. Oliveira introduced the idea of using a simple layer potential on an auxiliary boundary, which has been further explored. The desingularization technique, moving observation points away from the boundary, is also used to solve problems in inviscid flow, biharmonic problems, and spherical shell problems. In the case of Laplace's equation, the solution is expressed as an integral over an auxiliary boundary using a simple layer potential. This method is effective for solving elliptic boundary value problems and has been widely applied in various fields.