Transient heat transfer problems have found many applications to plenty of engineering and science problems. Most of the transient heat conduction problems have been numerically solved by meshless methods such as the element free Galerkin (EFG) method, reproducing kernel particle method (RKPM), meshless local Petrov-Galerkin (MLPG) and boundary element-free method (BFEM). Among all the meshless methods, the MLPG method has been widely used in solving heat conduction problems. But there exists two shortcomings which are the shape functions based on the moving least square (MLS) approximation lack the Kronecker’s delta property and numerical integration over the complicated MLS shape functions leads to an expensive numerical scheme. In order to eliminate these shortcomings of the MLS shape functions, the moving Kriging (MK) interpolation technique and the generalized MLS (GMLS) method can be employed instead of the traditional MLS approximation. So, some improved meshless local Petrov-Galerkin methods such as the direct MLPG (DMLPG) method have been established. Gu has firstly proposed the moving Kriging interpolation and has successfully developed a new moving Kriging interpolation-based EFG for solving one-dimensional steady-state heat conduction problems. Later, the meshless methods based on the moving Kriging interpolation have also been developed and studied, for example, the element free Galerkin method using moving Kriging interpolation, the meshless local Kriging (loKriging) method, and the boundary node method based on the moving Kriging interpolation. The DMLPG method, originated by Mirzaei and Schaback has been successfully applied to heat conduction, two and three dimensional potential problems, and anisotropic diffusion problems. Besides, efficiencies of the method for handling the elasticity problems must be mentioned. In this thesis, some preliminaries have been prepared and some materials about the moving Kriging interpolation, GMLS approximation and MLS approximation have been represented. One of the improved MLPG uses the moving Kriging interpolation techniques to construct meshless shape functions (for a set of randomly distributed points) which possess Kronecker’s delta property and therefore essential boundary conditions can be implemented easily. The local weak form of partial differential equations (PDEs) is derived by the weighted residual formulation based on a simple shaped local domain. Additionally, in implementation of the local weak form, the Heaviside step function is used as the test function. In another improved MLPG method -DMLPG method- boundary conditions are directly imposed, a GMLS method is used and numerical integrations involved in the local weak forms are shifted to on the polynomial basis functions instead of on the MLS shape functions. Then, these improved methods have been extended to the transient heat conduction problems and three numerical examples solved by them. In these examples, both regular and Halton points are employed. These methods applied to some fractional differential equations as well. A numerical example has been solved by the first improved MLPG method and reported its efficiencies compared to the existing schemes. Finally, we present some numerical results with both regular and scattered quasi-uniform mesh points to demonstrate the efficiencies of the DMLPG method in comparison with the ltr"