This is a short survey on the connection between general extension
theories and the study of realizations of elliptic operators A on smooth domains
in R^n, n >1. The theory of pseudodifferential boundary problems has turned
out to be very useful here, not only as a formulational framework, but also
for the solution of specific questions. We recall some elements of that theory,
and show its application in several cases (including new results), namely to
the lower boundedness question, and the question of spectral asymptotics for
differences between resolvents.