  We define the Eulerian ideal of a $k$-uniform hypergraph and study its degree
and Castelnuovo--Mumford regularity. The main tool is a Gr\"obner basis of the
ideal obtained combinatorially from the hypergraph. We define the notion of
parity join in a hypergraph and show that the regularity of the Eulerian ideal
is equal to the maximum cardinality of such a set of edges. The formula for the
degree involves the cardinality of the set of sets of vertices, $T$, that admit
a $T$-join. We compute the degree and regularity explicity in the cases of a
complete $k$-partite hypergraph and a complete hypergraph of rank $3$.
