  We develop a unified theory of Eulerian spaces by combining the combinatorial
theory of infinite, locally finite Eulerian graphs as introduced by Diestel and
K\"uhn with the topological theory of Eulerian continua defined as irreducible
images of the circle, as proposed by Bula, Nikiel and Tymchatyn.
  First, we clarify the notion of an Eulerian space and establish that all
competing definitions in the literature are in fact equivalent. Next,
responding to an unsolved problem of Treybig and Ward from 1981, we formulate a
combinatorial conjecture for characterising the Eulerian spaces, in a manner
that naturally extends the characterisation for finite Eulerian graphs.
Finally, we present far-reaching results in support of our conjecture which
together subsume and extend all known results about the Eulerianity of infinite
graphs and continua to date. In particular, we characterise all one-dimensional
Eulerian spaces.
