This talk aims to give an introduction to discontinuous Petrov-Galerkin (DPG) finite element methods and some of their applications. These high-order methods are especially crafted to be numerically stable by implicitly computing a near-optimal discrete broken (or discontinuous) test space as a function of a discrete trial space. The methods are extremely general in the sense that they can be applied to a variety of variational formulations associated to a given linear partial differential equation (PDE). These properties have mostly been harnessed to develop numerical methods that provide robust control of relevant equation parameters, like those in convection-diffusion equations and other singularly perturbed problems. Here, we show other interesting applications of DPG methods, including the development of finite element methods capable of solving PDEs in meshes with arbitrary polygonal element shapes and with improved conditioning properties (the growth of the condition number is O(h^-1), instead of the typical O(h^-2), where h is the mesh size). Additionally, their natural a posteriori error indicator has been used to solve problems of physical relevance, like the validation of dynamic mechanical analysis (DMA) calibration experiments of viscoelastic materials, and the modeling of form-wound medium-voltage stator coils sitting inside large electric machinery.