One-dimensional hybrid numerical simulations of tangential discontinuities (TDs) are presented for parameters representative of both the solar wind and magnetopause. For parameters common at the frontside magnetopause near local noon (antiparallel magnetic field and density gradients) it is found that TDs have a number of characteristic features that could be detected at the magnetopause. There is an overshoot in the magnetic field intensity on the low-density side of the TD. The overshoot is of the order of 20% for density ratios of 10 but can be somewhat larger for other parameters. It extends typically over a few ion Larmor radii. The various plasma quantities all vary on their own characteristic length scale, so that a given set of conditions on either side should result in a (possibly) unique structure. It is found that they key parameters in determining TD structure are the density (and ion temperature) ratios, the plasma beta on the magnetosheath side, and the value of the electric potential difference across the TD. The relative orientation of the magnetic fields on either side of the TD does not appear to be of critical importance. For typical solar wind conditions (density and field gradients parallel) the magnetic field profile does not posses an overshoot but it does extend over 2--3 ion Larmor radii. The presence of a cross-field plasma flow is shown to slightly modify the magnetic field profile, but the velocity profile is very sensitive to the sign of the flow (and hence the motional electric field). It is also shown that the TD structure is dependent on the assumed initial conditions. When the TD is initially chosen to be wider than the width that would result from the evolution of an infinitesimally thin initial TD, there is little further evolution. The currents in the TD layer can be a considerable fraction of the ion thermal speed, so that relaxation via plasma microinstabilities could arise. A detailed comparison with observed magnetopause crossing will be presented in a future article. ¿ American Geophysical Union 1991