  We introduce a class of overdetermined systems of partial differential
equations of finite type on (pseudo)-Riemannian manifolds that we call the
generalised Ricci soliton equations. These equations depend on three real
parameters. For special values of the parameters they specialise to various
important classes of equations in differential geometry. Among them there are:
the Ricci soliton equations, the vacuum near-horizon geometry equations in
general relativity, special cases of Einstein-Weyl equations and their
projective counterparts, equations for homotheties and Killing's equation.
  We also prolong the generalised Ricci soliton equations and, by computing
differential constraints, we find a number of necessary conditions for a
(pseudo)-Riemannian manifold $(M, g)$ to locally admit non-trivial solutions to
the generalised Ricci soliton equations in dimensions 2 and 3.
  The paper provides also a collection of explicit examples of generalised
Ricci solitons in dimensions 2 and 3 (in some cases).
