The finite element method (FEM) is a fundamental tool of the numerical solution of real-life problems based on partial differential equations. In the recent decades, various generalizations of the standard FEM have been developed. A lot of such extensions have been motivated by difficulties, arising in physical or engineering problems, that may be cumbersome to overcome with standard FEM techniques. Such situations are the presence of boundary layers, singularities or discontinuities in the solution, complex and/or evolving geometry of the physical domains etc. The tools of extension of the FEM may be enriching the polynomial approximation space with non-polynomial shape functions, allowing general polygonal/polyhedral cells, or use a boundary-unfitted mesh and restricted shape functions (either to a bulk domain or to a surface). This survey type talk gives a brief introduction to the main ideas of some generalized FEMs that use the above ideas: XFEM, VEM, CutFEM and TraceFEM.