The Eigenvalue Complementarity Problem (EiCP) finds important applications in different areas of science and engineering and differs from the traditional Eigenvalue Problem on the existence of nonnegative constraints on its variables and complementarity constraints between pairs of variables. In this talk the EiCP is first introduced together with some of its extensions and most important applications. The symmetric case is next considered and assumes that all the matrices involved in the definition of the EiCP are symmetric. The symmetric EiCP reduces to the problem of finding a stationary point of an appropriate nonlinear merit function on the simplex. A projected-gradient algorithm is recommended to deal with the symmetric EiCP by exploiting this nonlinear programming formulation. An enumerative algorithm is introduced to deal with the asymmetric EiCP. The method looks for a global minimum of an appropriate nonlinear program and requires in each node the computation of stationary points for this program. Computational experience for the solution of EiCPs is reported to highlight the efficiency of the projected-gradient and enumerative algorithms in practice.