  This paper is concerned with "nice" compactifications of manifolds.
Siebenmann's iconic dissertation characterized open manifolds M^m (m>5)
compactifiable by addition of a manifold boundary. His theorem extends easily
to cases where M^m is noncompact with compact boundary; however, when Bd(M^m)
is noncompact, the situation is more complicated. The goal becomes a
"completion" of M^m, ie, a compact manifold C^m and a compact subset A such
that C^m\A = M^m. Siebenmann did some initial work on this topic, and O'Brien
extended that work to an important special case. But, until now, a complete
characterization had yet to emerge. We provide such a characterization.
  Our second main theorem involves Z-compactifications. An open question asks
whether a well-known set of conditions laid out by Chapman and Siebenmann
guarantee Z-compactifiability for a manifold M^m. We cannot answer that
question, but we do show that those conditions are satisfied if and only if M x
[0,1] is Z-compactifiable. A key ingredient is the above Manifold Completion
Theorem---an application that partly explains our current interest in that
topic, and also illustrates the utility of the conditions found in that
theorem.
