In this report, the stability of the moving least squares (MLS) approximation is analyzed. It is shown that the stability deteriorates severely as the nodal spacing decreases. Then, a stabilized MLS approximation is developed. The stability and error estimates of the stabilized MLS approximation are derived theoretically and verified numerically. The stabilized MLS approximation is further introduced into the element-free Galerkin (EFG) method to produce a stabilized EFG method. Error analysis of the stabilized EFG method is provided for boundary value problems with mixed boundary conditions. Numerical examples are finally given to demonstrate the theoretical results.