The hydrodynamic basis for the accumulation of spermatozoa at surfaces has been investigated. The general conclusion is that when spermatozoa arrive at a surface, they will remain there if the vector of the time-averaged thrust is directed towards that surface. This can arise in two basic ways. First, consider spermatozoa that maintain a three-dimensional waveform and roll (spin) as they progress: in this case, it is argued that the conical (rather than cylindrical) shape of the flagellar envelope will establish the direction-of-thrust necessary for capture by the surface. (Additional findings, for spermatozoa of this type, are that the swim-trajectory is curved and that the direction of its curvature reveals the roll-direction of the cell.) Second, consider spermatozoa that maintain a strictly two-dimensional waveform at the surface: in this case, spermatozoa can be captured because the plane-of-flattening of the sperm head is tilted slightly relative to the plane of the flagellar beat. The sperm head is acting as a hydrofoil and, in one orientation only, it comes to exert a pressure against the surface. (This pressure may possibly, in vivo, aid the penetration of the zona pellucida.) The hydrofoil action of sperm heads may explain any bias in the circling direction of spermatozoa that execute two-dimensional waves at surfaces. Finally, a more complex phenomenon is where interaction of the spermatozoa with the surface appears to induce a three-dimensional to two-dimensional conversion of the flagellar wave (thus permitting the hydrofoil effect described). This is characteristic of sperm with 'twisted planar' rather than helical waves. In mammalian spermatozoa, approximately half the beat cycle is planar and the other half generates a pattern of torque causing the head to roll clockwise (seen from ahead), producing a torsion of the neck region of the flagellum. It is the gradual suppression of this torsion, by either impedance at the solid boundary or by raised viscosity, that converts the 'twisted planar' shape into a planar wave.