We present a collection of 52 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of real-life applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems according to their structural properties. Identifiers based on these and other properties can be used to extract particular types of problems from the collection. A brief description of each problem is given. NLEVP serves both to illustrate the tremendous variety of applications of nonlinear eigenvalue problems and to provide representative problems for testing, tuning, and benchmarking of algorithms and codes.