Main shortcoming of traditional methods such as finite element and finite volume is time consuming of grid generation process which takes more than 70 percent of total time. At the end of 20th century, meshless methods were developed and used readily. The main idea of these methods is approximation of unknown function in the form of a linear combination of functions which called shape functions and they don't need to grid in their domain. One of the famous meshless methods is called Meshless Local Petrov -Galerkin (MLPG). This method has been used widely by Atluri and Zhu since 1998 and recently has been achieved success by several modifications. In this project we deal three methods of function approximation with and then explain the method of MLPG. Furthermore we study several done modifications on the method. These modifications are · eliminating integration by applying Dirac delta function as test function (MLPG2) · eliminating integration on domain by using Heavy side (step) function as test function (MLPG5) · combination of simple methods in order to improve solutions on domain and boundary · applying some modification in integrating methods