Model order reduction is an integral approach to solving high-dimensional systems of ordinary and partial differential equations. As the needs for processing large amounts of data, resulting in high-order models, are only increasing, this approach will understandably stay relevant. Reduced basis methods are discussed, with the focus being on the proper orthogonal decomposition (POD). Their applications to parametric PDEs are presented as well. The encountered issue in this approach involves nonlinear systems, which are instead tackled by a combination of POD and the discrete empirical interpolation method (DEIM). Alongside them, newer versions of the DEIM algorithm are presented, namely the Q-DEIM and K-DEIM, which consist of utilizing the QR decomposition and clustering methods, respectively. Numerical experiments have shown the effectiveness of these approaches, to both synthetic and real-life problems, notably the FKPP equation, which is very important in biology. The potentially surprising results indicate that models whose dimensions are in hundreds or thousands may be accurately represented by systems of only several equations. However, the methods presented here encompass only a small part of the plethora of model order reduction methods