We consider several models of the damped oscillators in nonrelativistic quantum me-chanics in a framework of a general approach to the dynamics of the time-dependent Schr•odinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge trans-formations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time evolution of the expectation values of the energy-related operators is determined for two models of the quantum damped oscillators under con-sideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.