Almost resilient function is the generalization of resilient function and has important applications in multiple authenticate codes and almost security cryptographic Boolean functions. In this paper, some constructions are provided. In particular, the Theorem 3 in [7] is improved. As e-almost (n,1,k)-resilient functions play an important role in the secondary constructions, we concluded some properties and constructions. Specially we presented a spectral characterization of almost (n,1,k)-resilient functions, which can be used to identify an almost (n,1,k)-resilient function by computing its walsh spectra.