In this talk, I begin with a review of several mathematical models for describing wave motion in quantum and plasma physics. Computational difficulties for simulating wave propagation and interaction in quantum and plasma physics are discussed. Efficient and accurate numerical algorithms for computing ground and excited states as well as the dynamics of the nonlinear Schrodinger equation are presented. Applications in collapse and explosion of Bose-Einstein condensates, transport of ultrocold atoms in optical lattices, quantized vortics in superfluididty and wave interaction in plasma physics are reported. Finally, some conclusions are drawn.