  The aim of this paper is to compare and contrast the class of residually
finite groups with the class of equationally Noetherian groups - groups over
which every system of coefficient-free equations is equivalent to a finite
subsystem. It is easy to construct groups that are residually finite but not
equationally Noetherian (e.g. the direct sum of all finite groups) or vice
versa (e.g. the additive group $(\mathbb{Q},+)$ of the rationals). However, no
explicit examples that are finitely generated seem to appear in the literature.
  In this paper, we show that among finitely generated groups, the classes of
residually finite and equationally Noetherian groups are similar, but neither
of them contains the other. On one hand, we show that some classes of finitely
generated groups which are known to be residually finite, such as
abelian-by-polycyclic groups, are also equationally Noetherian (answering a
question posed by R. Bryant). We also give analogous results stating sufficient
conditions for a fundamental group of a graph of groups to be equationally
Noetherian and to be residually finite. On the other hand, we produce examples
of finitely generated non-(equationally Noetherian) groups which are either
residually torsion-free nilpotent or conjugacy separable, as well as examples
of finitely presented equationally Noetherian groups that are not residually
finite.
