I am reading some basic context books about topology (i.e. The Poincaré Conjecture, by Donal O'Shea between others) and following this open Topology and Geometry video lectures of the brilliant professor Tadashi Tokieda in the African Institute for Mathematical Sciences (for beginners in the matter, if you have time I would suggest you to have a look to them!).
I would like to ask the following question:
Is every $n$-manifold the boundary of an $(n+1)$-manifold? Is every compact $n$-manifold the boundary of a compact $(n+1)$-manifold?
p.s. This question was rewritten according to the suggestions in the comments and Meta (here), I hope now will be more accurate. Thanks to everybody for the suggestions!