We study the laminar boundary-layer flow over a general spheroid rotating in otherwise still fluid. In particular, we distinguish between prolate and oblate spheroids and use an appropriate spheroidal coordinate system in each case. An eccentricity parameter e is used to distinguish particular bodies within the oblate or prolate families and the laminar-flow equations are established for each family with e as a parameter. In each case, setting e = 0 reduces the equations to those already established in the literature for the rotating sphere. We begin by solving the laminar-flow equations at each latitude using a series-solution approximation. A comparison is then made to solutions obtained from an accurate numerical method. The two solutions are found to agree well for a large range of latitudes and eccentricities, and at these locations the series solution is to be preferred due to its simplicity and ease of computation. A discussion of the resulting flows is given with particular emphasis on the implications for their hydrodynamic stability. Their stability characteristics are expected to be very similar to those over the rotating sphere as already studied in the literature.