We consider in this paper two graph orientation problems. The input of both problems is (i) a mixed graph G whose vertex set is V and edge set (resp. arc set) is E (resp. A) and (ii) a set P V V of source-target pairs. The first problem, called S-GO, is a decision problem introduced by Hassin and Megiddo (Linear Algebra and its Applications 114 (1989): 589-602) and defined as follows: is it possible to find an orientation of G that replaces each edge (u; v) 2 E by a single arc (either uv or vu) in such a way that, for each (s; t) 2 P, there exists a directed path from s to t ? Our second problem, called MIN-D-GO, is a minimization problem that can be seen as a variant of S-GO, in which we allow some edges (u; v) 2 E to be doubly oriented. The goal is then to find an orientation of G that replaces each edge (u; v) 2 E by uv and/or vu in such a way that (i) there exists a directed path from s to t for each (s; t) 2 P and (ii) the number of doubly oriented edges is minimized. We investigate the complexity of SGO and MIN-D-GO by considering some restrictions on the input instances (such as the maximum degree of G or the cardinality of P). We provide several polynomial time algorithms, hardness and inapproximability results that together give an extensive picture of tractable and intractable instances for both problems.