This textbook provides an accessible and self-contained description of the Galerkin finite element method for the two important models of continuum mechanics, transient heat conduction and elastodynamics, from formulation of the governing equations to implementation in Matlab. The coverage follows an intuitive approach: the salient features of each initial boundary value problem are reviewed, including a thorough description of the boundary conditions; the method of weighted residuals is applied to derive the discrete equations; and clear examples are introduced to illustrate the method Introduction -- Mesh free methods for engineering problems -- Mechanics of solids and structures -- Principles for weak forms -- MFree shape function construction -- Element free Galerkin method -- Meshless local Petrov-Galerkin method -- Point interpolation methods -- Mesh free methods for fluid dynamics problems -- Mesh free methods for beams -- Mesh free methods for plates -- Mesh free methods for shells -- Boundary mesh free methods -- Mesh free methods coupled with other methods -- Implementation issues -- MFree2D