In this talk, we present $H^1$-, H(div)- and H(curl)-conforming spectral method with exact preservation of the curl/divergence-free constraints for two typical PDEs arising from plasma simulations. One is the Vlasov-Ampere system and the other one is incompressible visco-resistive MHD system. Two key ingredients, i.e. exact de Rham complexes and their commuting diagram, and the derivative property of the generalized Jacobi polynomials, are essential for the derivation of the desired basis functions. Fast matrix-free solution algorithm particularly designed for scalable and parallel computations are proposed. Besides, we present efficient asymptotic-preserving schemes, which guarantee that the asymptotic limiting of the discrete scheme is a consistent and stable discretization of the quasi-neutral limit of the continuous model. Ample numerical examples in 2D and 3D illustrate both the accuracy and efficiency of the proposed methods.