  The celebrated Cwikel-Lieb_Rozenblum inequality gives an upper estimate for
the number of negative eigenvalues of Schroedinger operators in dimension three
and higher. The situation is much more difficult in the two dimensional case.
There has been significant progress in obtaining upper estimates for the number
of negative eigenvalues of two dimensional Schroedinger operators on the whole
plane. In this thesis, we present estimates of the Cwikel-Lieb_Rozenblum type
for the number of eigenvalues (counted with multiplicities) of two dimensional
Schroedinger operators lying below the essential spectrum in terms of the norms
of the potential. The problem is considered on the whole plane with different
supports of the potential of dimension between 0 and 2, and on a strip with
various boundary conditions. In both cases, the estimates involve weighted L^1
norms and Orlicz norms of the potential.
