``The mechanism for generating the geomagnetic field remains one of the central unsolved problems in geoscience.'' So states the report on the National Geomagnetic Initiative (NGI) prepared by the U.S. Geodynamics Committee, et al. [1993], with advice from the NGI Workshop held in Washington D.C. in March 1992. All analyses of the geomagnetic data point to the core as containing the source of the field and ``The basic premise that virtually everyone accepts is that the Earth's magnetism is created by a self-sustaining dynamo driven by fluid motions in Earth's core'' (NGI, p.135). Dynamical questions at once arise, such as ``What is the energy source driving those motions?'' Jacobs [1953] proposed that the solid inner core (SIC) is the result of the freezing of the fluid outer core (FOC). Verhoogen [1961] noticed that the release of latent heat at the inner core boundary (ICB) during freezing would help drive thermal convection in the FOC, and Braginsky [1963] pointed out that the release of the light alloying elements during fractionation at the ICB would provide compositional buoyancy. These two sources suffice to supply the geodynamo with energy throughout geological time, even in the absence of dissolved radioactivity in the core [ Braginsky and Roberts, 1994a; Kuang et al., 1994]. Stevenson [1991] argues that potential differences on the core--mantle boundary (CMB) of electrochemical origin may be partially responsible for the geomagnetic field.
Whatever the driving mechanism, it is clear that the magnetohydrodynamics (MHD) of the core must be understood. This has proved to be a challenging task; progress has been slow. The directional property of the magnetic compass needle demonstrates that Coriolis forces play an essential role. Because the molecular diffusivities of heat and composition are so small, these sources of buoyancy must instead be transported across the core by turbulence. The phenomena of interest arise from slight deviations in the FOC from a well--mixed adiabatic state, and theory must consistently disentangle these from the large ``background.'' In short, it is far from obvious what equations best describe large scale core MHD [ Braginsky and Roberts, 1994a].
We shall concentrate below on the research of US scientists, even when it was carried out abroad.
We shall add the work of foreign scientists who have long standing affiliations with US Institutions.
US theoreticians have played a major role in elucidating fast dynamos, which amplify fields on the same time scales as the flows.
In contrast, slow dynamos act on a diffusive timescale, based on the magnetic diffusivity, 
, of the conductor.
Interestingly, McFadden and Merrill [1993] have recently derived
m
s
from the paleomagnetic data,
but we shall take 
3 m
s
.
The diffusive time scale of the core is therefore about 10
years, and the geodynamo
problem is to understand how the geomagnetic field is maintained over times that are
substantially longer than this.
Fast dynamo theory has no obvious contributions to make, and will therefore not be
considered here. The core is a slow dynamo.