Here, we will review as well as will present some new developments of the optimal local truncation error method (OLTEM) for the solution of partial differential equations. Similar to the finite difference method, the structure and the width of discrete equations are assumed in advance. The unknown coefficients of the discrete system are calculated by the minimization of the order of the local truncation error. The main advantages of OLTEM are an optimal high accuracy of discrete equations and the simplicity of the formation of a discrete (semi-discrete) system for irregular domains and interfaces (composite materials). In contrast to finite elements, trivial unfitted Cartesian meshes (no need in complicated mesh generators) are used with OLTEM. The known interface and boundary conditions at small number of selected points are added to the discrete equations as the right-hand side. A new OLTEM post-processing procedure for the calculation of the spatial derivatives of the primary functions (e.g., stresses or heat fluxes) that is based on the use of original PDEs significantly increases the accuracy of the spatial derivatives. For example, we have obtained the 10-th order of accuracy for stresses calculated by OLTEM with ‘quadratic elements’ applied to elastostatics problems with heterogeneous materials and irregular interfaces. Currently, OLTEM has been applied to the solution of the wave, heat, elastodynamics, Helmholtz, Poisson, Stoke’s and elastostatics equations. The theoretical and numerical results show that at the computational costs of linear finite elements, OLTEM yields the 4th order of accuracy for the considered scalar PDEs on irregular domains (it is much more accurate compared with linear and high-order finite elements at the same number of degrees of freedom). E.g., 3-D numerical examples on irregular domains show that at accuracy of 5%, OLTEM applied to the wave equation reduces the number of degrees of freedom by a factor of greater than 1000 compared to that for linear finite elements. At the computational costs of quadratic finite elements, OLTEM yields the 10-th order of accuracy for the time-independent elasticity equations and the 11-th order of accuracy for the Poisson equation with complex irregular interface, i.e., the increase in accuracy by 7 and 8 orders compared to FEM.