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Mariner 10 measurements proved the existence of a large-scale internal magnetic field on Mercury. The observed field amplitude, however, is too weak to be compatible with typical convective planetary dynamos. The Lorentz force based on an extrapolation of Mariner 10 data to the dynamo region is 10−4 times smaller than the Coriolis force. This is at odds with the idea that planetary dynamos are thought to work in the so-called magnetostrophic regime, where Coriolis force and Lorentz force should be of comparable magnitude. Recent convective dynamo simulations reviewed here seem to resolve this caveat. We show that the available convective power indeed suffices to drive a magnetostrophic dynamo even when the heat flow though Mercury's core-mantle boundary is subadiabatic, as suggested by thermal evolution models. Two possible causes are analyzed that could explain why the observations do not reflect a stronger internal field. First, toroidal magnetic fields can be strong but are confined to the conductive core, and second, the observations do not resolve potentially strong small-scale contributions. We review different dynamo simulations that promote either or both effects by (1) strongly driving convection, (2) assuming a particularly small inner core, or (3) assuming a very large inner core. These models still fall somewhat short of explaining the low amplitude of Mariner 10 observations, but the incorporation of an additional effect helps to reach this goal: The subadiabatic heat flow through Mercury's core-mantle boundary may cause the outer part of the core to be stably stratified, which would largely exclude convective motions in this region. The magnetic field, which is small scale, strong, and very time dependent in the lower convective part of the core, must diffuse through the stagnant layer. Here, the electromagnetic skin effect filters out the more rapidly varying high-order contributions and mainly leaves behind the weaker and slower varying dipole and quadrupole components (Christensen in Nature 444:1056-1058, 2006). Messenger and BepiColombo data will allow us to discriminate between the various models in terms of the magnetic fields spatial structure, its degree of axisymmetry, and its secular variation. [ABSTRACT FROM AUTHOR]