Magnetic reconnection in a plane current sheet is investigated in both two and three dimensions, using three different types of simulation codes, Hall MHD, hybrid (electron fluid, kinetic ions), and a new code called Hall-less hybrid. The latter code, which is similar to the hybrid code but has the Hall term removed, enables us to clarify the differences between kinetic ion and Hall MHD approaches. The major findings of this research are (1) Sweet-Parker regime of reconnection cannot be maintained and does not reach a steady state in a kinetic plasma for physically interesting parameter regimes. (2) Fast asymptotic reconnection rate of 0.15VA0B0 is obtained both in the hybrid and Hall-less hybrid simulations with outflow boundaries. VA0 and B0 are the Alfv¿n velocity and magnetic field strength in the upstream region. This finding has two immediate implications. First, ion kinetics are sufficient to lead to fast reconnection even in the absence of the Hall term, and second, explanation of fast reconnection in terms of quadratic disperion of whistlers needs to be reconsidered, as whistlers are dispersionless in Hall-less hybrid limit. (3) Unlike in MHD, diffusion region is different in size that the region of localized resistivity. (4) While both Hall and hybrid codes show that reconnection is inherently asymmetric in three dimensions, there are differences in the nature of the asymmetry. In Hall MHD we show that the X-line grows in the direction of the electron drift, propagating as a (reconnection) wave because the current is carried by electrons, although the wave direction can change in the presence of a substantial ion flow. However, in the hybrid simulations here, as is the case for typical conditions at the magnetopause and magnetotail, ions carry the bulk of the current, and the observed asymmetry is found to be due to ion flow and not a wave motion unless the extent of finite resistivity in the third dimension is very thin, comparable to the current sheet thickness. Thus aside from scenarios where electrons are the dominant current carriers, such as very thin current sheets that are on electron scales, we do not expect the reconnection wave to form. This result is relevant to the magnetotail where dawn-dusk asymmetries are observed in the motions of auroral brightenings and surges, as well as in the statistical location of pressure decreases, flows, and magnetic signatures associated with the near-Earth neutral line and early plasmoids. We attribute the observed dawn-dusk asymmetries to ion flows. One interesting question left for future work is the possibility that reconnection waves may form in thin electron-scale current layers that are sometimes observed embedded within a thicker sheet in the magnetotail.