The physics of magnetic reconnection at Earth's dayside magnetopause is addressed. We present data from three-dimensional resistive magnetohydrodynamics (MHD) simulations, demonstrating that when the plasma resistivity is spatially uniform and constant in time, magnetic reconnection occurs at the subsolar point via the flux pileup mechanism. When the interplanetary magnetic field (IMF) is due south, the dependence of the magnetic pileup on the Lundquist number is consistent with that predicted by simple analytical solutions of the resistive MHD equations (see, for example, Sonnerup and Priest, J. Plasma Phys., 14, 283, 1975). Thus, one expects the reconnection rate in such resistive MHD models to stall at a critical value of the Lundquist number, making it unlikely that such models will reproduce Dungey's open magnetosphere in the high Lundquist number limit (i.e., Petschek's slow shock mechanism does not appear to be relevant in such models). We present a possible solution of this flux pileup saturation problem: if the spatial scale of the stagnation point flow is comparable to the ion inertial length, then Hall electric fields can permit the magnetic flux pileup to saturate before the reconnection begins to stall. If time permits, we will discuss -- in the context of resistive MHD simulations -- the dependence of the geometry of dayside magnetopause reconnection on the IMF orientation.