Let Cay(S:H) be the Cayley digraph of the generators S in the group H. A one-way infinite Hamiltonian path in the digraph G is a listing of all the vertices \([v_ i: 1\leq i<\infty]\), such that there is an arc from \(v_ i\) to \(v_{i+1}\). A two-way Hamiltonian path is defined similarly, with i ranging from -\(\infty\) to \(\infty\). In this paper, conditions on S and H for the existence of one- and two-way infinite Hamiltonian paths in Cay(S:H) are presented. If S is countably infinite and H is abelian, then Cay(S:H) has one- and two-way Hamiltonian paths if and only if it is strongly connected (except for one infinite family). Necessary and sufficient conditions on S are also given for Cay(S:H) to be strongly connected for a large class of Cayley digraphs. It is shown that any Cayley digraph of a countable locally finite group has both one- and two- way infinite Hamiltonian paths. A relation between strong connectivity and the outer valence of finite vertex-transitive digraphs is also presented.