  The present review will focus on recent development of exact-diagonali-
zation (ED) methods that use Lanczos algorithm to transform large sparse
matrices onto the tridiagonal form. We begin with a review of basic principles
of the Lanczos method for computing ground-state static as well as dynamical
properties. Next, generalization to finite-temperatures in the form of well
established finite-temperature Lanczos method is described. The latter allows
for the evaluation of temperatures T>0 static and dynamic quantities within
various correlated models. Several extensions and modification of the latter
method introduced more recently are analysed. In particular, the
low-temperature Lanczos method and the microcanonical Lanczos method,
especially applicable within the high-T regime. In order to overcome the
problems of exponentially growing Hilbert spaces that prevent ED calculations
on larger lattices, different approaches based on Lanczos diagonalization
within the reduced basis have been developed. In this context, recently
developed method based on ED within a limited functional space is reviewed.
Finally, we briefly discuss the real-time evolution of correlated systems far
from equilibrium, which can be simulated using the ED and Lanczos-based
methods, as well as approaches based on the diagonalization in a reduced basis.
