  Given a metrically complete Riemannian manifold $(M,g)$ with smooth nonempty
boundary and assuming that one of its curvatures is subject to a certain bound,
we address the problem of whether it is possibile to realize $(M,g)$ as a
domain inside a geodesically complete Riemannian manifold $(M',g')$ without
boundary, by preserving the same curvature bounds. In this direction we provide
three kind of results: (1) a general existence theorem showing that it is
always possible to obtain a geodesically complete Riemannian extension without
curvature constraints; (2) various topological obstructions to the existence of
a complete Riemannian extension with prescribed sectional and Ricci curvature
bounds; (3) some existence results of complete Riemannian extensions with
sectional and Ricci curvature bounds, mostly in the presence of a convexity
condition on the boundary.
