The meshless local Petrov-Galerkin (MLPG) method is an effective meshless method to solve partial differential equations. In this article, the MLPG method is used to solve nonlinear steady and transient heat conduction problems. The essential boundary condition is enforced by the method of direct interpolation. The moving least-squares (MLS) method is used for interpolation. Thermal conductivity of the material is assumed to be dependent on the temperature. An iterative procedure based on the predictor-corrector method is used. Time integration is performed using the method. Results are compared with the available exact solution and the solution by the finite-element method, and is found to be in good agreement.