A topological group is said to be locally pseudocompact if the identity has a pseudocompact neighborhood (equivalently: if the identity has a local basis of pseudocompact neighborhoods). Such groups are locally bounded in the sense of A. Weil, so each such group G is densely embedded in an essentially unique locally compact group G (called its Weil completion). The authors present necessary and sufficient conditions of local and global nature for a locally bounded group to be locally pseudocompact, as follows. Theorem. If G is a locally bounded group with Weil completion G, then the following conditions are equivalent: (i) G is locally pseudocompact; (ii) G is C * -embedded in G (i.e., βG = βG); (iii)G is C-embedded in G (i.e., υG = υG); (iv) G is M-embedded in G (i.e., γG = G); (v) some nonempty open subset U of G satisfies β(cl G U) = cl G U; (vi) every bounded open subset U of G satisfies β(cl G U) = cl G U.