In this paper, we study the initial boundary value problem for the time-fractional Oldroyd-B fluid equation. Our model contains two Riemann–Liouville fractional derivatives, which have many applications, for example, in viscoelastic flows. For the linear case, we obtain regularity results under different assumptions on the initial data and the source function. For the nonlinear case, we establish the existence and uniqueness of a solution using Banach’s fixed point theorem.