  This survey/expository article covers a variety of topics related to the
"topology at infinity" of noncompact manifolds and complexes. In manifold
topology and geometric group theory, the most important noncompact spaces are
often contractible, so distinguishing one from another requires techniques
beyond the standard tools of algebraic topology. One approach uses end
invariants, such as the number of ends or the fundamental group at infinity;
another approach seeks nice compactifications, then analyzes the boundaries. A
thread connecting the two is shape theory.
  In these notes we provide a careful development of several topics: homotopy
and homology properties and invariants for ends of spaces, proper maps and
homotopy equivalences, tameness conditions, shapes of ends, and various types
of Z-compactifications and Z-boundaries. Classical and current research from
both manifold topology and geometric group theory provide the context. Along
the way, many open problems are encountered. Our primary goal is casual but
coherent introduction that is accessible to graduate students and also of
interest to active mathematicians whose research might benefit from knowledge
of these topics.
