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.