Summary: We deal with a new stochastic control problem for a class of uncertain discrete-time stochastic systems with randomly varying sensor delay. The parameter uncertainties are allowed to be time-varying and norm-bounded, and appear in the system state matrix. The system measurements are subject to randomly varying sensor delays, which often occur in information transmission through network. The aim of this paper is to design an output feedback controller such that, for all admissible parameter uncertainties and all probabilistic sensor delays, the closed-loop system is Bounded-Input Bounded-Output (BIBO) stable in the mean square (i.e. the system state is mean square bounded), and the steady-state variance of each state is not more than the individual prescribed upper bound. It is shown that such a controller design problem is feasible if there are positive definite solutions to a couple of algebraic Riccati-like inequalities or linear matrix inequalities. The explicit expression of the desired robust controllers is derived in terms of some free parameters. An illustrative numerical example is exploited to demonstrate the usefulness and flexibility of the proposed design approach.