We consider quasiperiodic and periodic (cnoidal) wave solutions of a set of n-component vector nonlinear Schrödinger equations (VNLSEs). In a biased photorefractive crystal with a drift mechanism of nonlinear response and Kerr-type nonlinearity, n-component nonlinear Schrödinger equations can be used to model self-trapped mutually incoherent wave packets. These equations also model pulse-pulse interactions in wavelength-division-multiplexed channels of optical fiber transmission systems. Quasiperiodic wave solutions for the VNLSEs in terms of n-dimensional Kleinian functions are presented. Periodic solutions in terms of Hermite polynomials and generalized Hermite polynomials for n-component nonlinear Schrödinger equations are found.