A review is presented of the unitarity problem in modern quantum cosmology - quantum gravity theory of spatially closed worlds. Based on the usual path-integral quantization of true physical variables, we arrive at the standard Dirac-Wheeler-DeWitt formulation of quantum cosmology, which now becomes supplemented with the algorithms for conserved probability amplitudes and thus constitutes a manifestly unitary field theory. The related issue of the many-fingered time parametrizing the unitary evolution in quantum cosmology is discussed in much detail. We also give a review of the generalized canonical quantization in relativistic gauges of Batalin, Fradkin and Vilkovisky (BFV) and show that the Dirac-Wheeler-DeWitt formalism occupies an intermediate position between quantizing the physical variables and the BFV theory. The unitarity approach, which is developed in the exact theory within the framework of the formal path-integral quantization, is confirmed at a strictly operatorial level in the one-loop approximation of a semi- classical loop expansion. In particular, the general semiclassical solution of the Wheeler-DeWitt equations is presented and its implications in the theory of the early inflationary Universe are considered. The prospects of the third-order quantization and the unification of unitary quantum cosmology with the Euclidean quantum gravity are briefly discussed.

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