Abstract: The lecture summarizes the habilitation thesis of the author presented at the University of Ostrava. It is based on a series of papers covering work from the period 2007-2015 on spectral properties of Schroedinger operators: these are problems which deal with spectral analysis of the Schroedinger operators with the potentials unbounded from below exhibiting spectral transitions, the operators defined on geometrically nontrivial regions, the magnetic Laplacians in three dimensional magnetic bottles, and periodic quantum graphs with gaps in the spectrum.