Abstract A very general random fixed point theorem is first proved. As applications, some random approximation theorems and random fixed-point theorems are obtained. Recent results due to Itoh [ J. Math. Anal. Appl. 67 (1979), 261-273], Lin [ Proc. Amer. Math. Soc. 103 (1988), 502-506], and Xu [ Proc. Amer. Math. Soc. 110 (1990), 495-500] either are improved or are direct consequences of our results. The stochastic versions of the recent results of Lin  and Li [ Proc. Amer. Math. Soc. 97 (1986), 277-280] are also given.