  The notion of an odd quasi-connection on a supermanifold, which is loosely an
affine connection that carries non-zero Grassmann parity, is examined. Their
torsion and curvature are defined, however, in general, they are not tensors. A
special class of such generalised connections, referred to as odd connections
in this paper, have torsion and curvature tensors. Part of the structure is an
odd involution of the tangent bundle of the supermanifold and this puts drastic
restrictions on the supermanifolds that admit odd connections. In particular,
they must have equal number of even and odd dimensions. Amongst other results,
we show that an odd connection is defined, up to an odd tensor field of type
$(1,2)$, by an affine connection and an odd endomorphism of the tangent bundle.
Thus, the theory of odd connections and affine connections are not completely
separate theories. As an example relevant to physics, it is shown that $N= 1$
super-Minkowski spacetime admits a natural odd connection.
