  Li\'{e}nard-type nonlinear oscillators with linear and nonlinear damping
terms exhibit diverse dynamical behavior in both the classical and quantum
regimes. In this paper, we consider examples of various one-dimensional
Li\'{e}nard type-I and type-II oscillators. The associated Euler-Lagrange
equations are divided into groups based on the characteristics of the damping
and forcing terms. The Li\'{e}nard type-I oscillators often display localized
solutions, isochronous and non-isochronous oscillations and are also precisely
solvable in quantum mechanics in general, where the ordering parameters play an
important role. These include Mathews-Lakshmanan and Higgs oscillators.
However, the classical solutions of some of the nonlinear oscillators are
expressed in terms of elliptic functions and have been found to be
quasi-exactly solvable in the quantum region. The three-dimensional
generalizations of these classical systems add more degrees of freedom, which
show complex dynamics. Their quantum equivalents are also explored in this
article. The isotonic generalizations of the non-isochronous nonlinear
oscillators have also been solved both classically and quantum mechanically to
advance the studies. The modified Emden equation categorized as Li\'{e}nard
type-II exhibits isochronous oscillations at the classical level. This property
makes it a valuable tool for studying the underlying nonlinear dynamics. The
study on the quantum counterpart of the system provides a deeper understanding
of the behavior in the quantum realm as a typical PT-symmetric system.
