  The paper contains an exposition of recent as well as old enough results on
determinantal random point fields. We start with some general theorems
including the proofs of the necessary and sufficient condition for the
existence of the determinantal random point field with Hermitian kernel and a
criterion for the weak convergence of its distribution. In the second section
we proceed with the examples of the determinantal random point fields from
Quantum Mechanics, Statistical Mechanics, Random Matrix Theory, Probability
Theory, Representation Theory and Ergodic Theory. In connection with the Theory
of Renewal Processes we characterize all determinantal random point fields in
R^1 and Z^1 with independent identically distributed spacings. In the third
section we study the translation invariant determinantal random point fields
and prove the mixing property of any multiplicity and the absolute continuity
of the spectra. In the fourth (and the last) section we discuss the proofs of
the Central Limit Theorem for the number of particles in the growing box and
the Functional Central Limit Theorem for the empirical distribution function of
spacings.
