In this paper, we first prove some random fixed point theorems for random nonexpansive operators in Banach spaces. As applications, some random approximation theorems for random 1-set-contraction or random continuous condensing mappings defined on closed balls of a separable Banach space, or on separable closed convex subsets of a Hilbert space or on spheres of infinite dimensional separable Banach spaces are established. Our results are generalizations, improvements or stochastic versions of the corresponding results of Bharucha-Reid (1976), Lin (1988, 1989), Lin and Yen (1988), Massatt (1983), Sehgal and Waters (1984) and Xu (1990).