We have studied 38 low-latitude, dayside (0800-1600 LT) magnetopause crossings by the AMPTE/IRM satellite to investigate the variations of key plasma parameters and the magnetic field in the magnetosheath region adjacent to the dayside magnetopause. We find that the structures of the key plasma parameters and the magnetic field and the dynamics of plasma flows in this region depend strongly on the magnetic shear across the magnetopause, that is, on the angle between the magnetosheath magnetic field and the geomagnetic field. When the magnetic shear is low (<30°), a magnetosheath transition layer, also called the “plasma depletion layer,” of 10-min average width exists where the magnetosheath magnetic field piles up against the magnetopause. In this region the plasma density and plasma β as well as the proton and electron temperatures are lower than in the magnetosheath proper. The condition for the onset of the mirror instability is generally not met in the magnetosheath transition layer, where the plasma β often falls below 1, while it is marginally satisfied in the magnetosheath proper, where usually β>1. When the magnetic shear across the magnetopause is high (>60°), the near-magnetopause magnetosheath is more disturbed. The magnetic field in this case does not pile up in the immediate vicinity of the magnetopause, and no systematic variations in the plasma parameters are observed in this region until the encounter of the magnetopause current layer; that is, there is no magnetosheath transition layer. Also in contrast to the low-shear case, the mirror instability threshold is marginally satisfied throughout the magnetosheath. The plasma flow pattern in the magnetosheath region adjacent to the dayside magnetopause is also found to depend strongly on the magnetic shear across the magnetopause: the magnetosheath flow component tangential to the magnetopause is enhanced and rotates to become more perpendicular to the local magnetic field as the low-shear magnetopause is approached. This flow behavior may be consistent with the formation of a stagnation line instead of a stagnation point at the subsolar magnetopause. Enhancement and rotation of the magnetosheath flow on approach to the magnetopause are rarely observed when the magnetic shear across the magnetopause is high. In essence, our observations provide evidence for high (low) rate of transfer of magnetic flux and mass across the magnetopause when the magnetic shear is high (low). The relationships between the electron and proton temperature anisotropies and β in the near-magnetopause magnetosheath region are also examined. It is found that Te⊥/Te∥ remains close to 1 for the entire range of βe, whereas Tp⊥/Tp∥ is generally anticorrelated with βp∥. However, no universal relationship seems to exist between Tp⊥/Tp∥ and βp∥.