  A central issue in the theory of astrophysical compact objects and heavy ion
reactions at intermediate and relativistic energies is the Nuclear Equation of
State (EoS). On one hand, the large and expanding set of experimental and
observational data is expected to constrain the behaviour of the nuclear EoS,
especially at density above saturation, where it is directly linked to
fundamental processes which can occur in dense matter. On the other hand,
theoretical predictions for the EoS at high density can be challenged by the
phenomenological findings. In this topical review paper we present the
many-body theory of nuclear matter as developed along different years and with
different methods. Only nucleonic degrees of freedom are considered. We compare
the different methods at formal level, as well as the final EoS calculated
within each one of the considered many-body schemes. The outcome of this
analysis should help in restricting the uncertainty of the theoretical
predictions for the nuclear EoS.
