  Study of fine spectral properties of quasiperiodic and similar discrete
Schr\"odinger operators involves dealing with problems caused by small
denominators, and until recently was only possible using perturbative methods,
requiring certain small parameters and complicated KAM-type schemes. We review
the recently developed nonperturbative methods for such study which lead to
stronger results and are significantly simpler. Numerous applications mainly
due to J. Bourgain, M. Goldstein, W. Schlag, and the author are also discussed.
