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In a complete bipartite graph \(K_{s,t}\), each vertex of one vertex set is joined to each vertex of the second vertex set by exactly one edge; An Eulerian orientation of \(K_{s,t}\) assigns directions to the edges in such a way that the resulting digraph has an Eulerian dicircuit. Similarly, any Eulerian circuit of \(K_{s,t}\) ascribes directions to the edges and results in an Eulerian orientation. This paper investigates Eulerian orientations and circuits of \(K_{s,t}\). Exact solutions are presented for \(s = 2\) and \(t = 4\). Computer searches were used to obtain results for other small values of \(s\) and \(t\). These results also lead to two conjectures which deal with upper and lower bounds on the numbers of Eulerian circuits.