Quantum effects in cosmology

A review of quantum effects in cosmology is given. The main emphasis is made on physical interpretations and possible observational consequences of the effects. The paper includes the following sections: 1. Introduction. The present state of the Universe. 2. What can we expect from a complete cosmological theory? 3. An overview of quantum effects in cosmology. 4. Parametric (superadiabatic) amplification of classical waves. 5. Graviton creation in the inflationary universe. 6. Quantum states of a harmonic oscillator. 7. Squeezed quantum states of relic gravitons and primordial density perturbations. 8. Quantum cosmology, minisuperspace models and inflation. 9. From the space of classical solutions to the space of wavefunctions. 10. On the probability of quantum tunneling from 'nothing'. 11. Duration of inflation and possible remnants of the preinflationary Universe. 12. Relic gravitons and the birth of the Universe.