In this paper, we consider constant step-size stochastic approximation with delayed updates. For the nondelayed case, it is well known that under appropriate conditions, the discrete-time iterates of stochastic approximation track the trajectory of a continuous-time ordinary differential equation (ODE). For the delayed case, we show in this paper that, under appropriate conditions, the discrete-time iterates track the trajectory of a delay-differential equation (DDE) rather than an ODE. Thus, delayed updates lead to a qualitative change in the behavior of constant step-size stochastic approximation. We present multiple examples to illustrate the qualitative affect of delay and show that increasing the delay is generally destabilizing but, for some systems, it can be stabilizing as well.