Recently, the study on tensor eigenvalue complementarity problem (TEiCP) has at-tracted much attention due to its various applications in polynomial optimization and differential dynamical systems. In this talk, we will briefly introduce the the general properties of TEiCP, including the solution existence and uniqueness. In particular, we show that there exists a unique solution of eigenvalue complementarity problem for irreducible nonnegative tensors. For the symmetric case, we derive a sufficient and nec-essary condition for the solvability of TEiCP by reformulating it as a nonlinear program. Based on the properties, we propose two numerical methods: shifted projected power method and damped semismooth Newton method. Some numerical experiments are also presented to show the efficiency of our methods.