This dissertation provides the foundation for an in-depth understanding and significant development of the tail-equivalent linearization method (TELM) to solve different classes of nonlinear random vibration problems. The TELM is a linearization method that uses the first-order reliability method (FORM) to define a tail-equivalent linear system (TELS) and to estimate the tail of the response distribution for nonlinear systems under stochastic inputs. The method was originally developed in the time domain for inelastic systems. It was later extended in the frequency domain for a specific class of nonlinear excitations, while the frequency domain version for inelastic systems is covered in the present work. This dissertation mathematically formalizes and extends TELM analysis with different types of discretization of the input process. A general formulation for discrete representation of a Gaussian band-limited, white-noise process is introduced, which employs the sum of deterministic and orthogonal basis functions weighted by random coefficients. The selection of the basis functions completely defines the two types of discretizations used in the earlier works. Specifically, a train of equally spaced time delta-Dirac functions leads to the current time-domain discretization, while harmonic functions with equally spaced frequencies lead to the current frequency-domain discretization. We show that other types of orthogonal basis functions can be used with advantage to represent a Gaussian band-limited white noise and in particular we employ sinc basis functions, which are at the base of the Whittaker-Shannon interpolation formula. We demonstrate that this representation is suitable for reducing the total number of random variables that are necessary to describe the process, since it decouples the computational-time discretization from the band-limit of the process. Next, the dissertation tackles the problem of a nonlinear system subjected to multi-component excitations by defining an augmented standard normal space composed of all the random variables that define the multiple components of the excitation. The tail-equivalent linearization and definition of the TELS is taken in this new space. Once the augmented TELS is defined, response statistics of interest are determined by linear random vibration analysis by superposition of responses due to each component of the excitation. The method is numerically examined for an asymmetric structure with varying eccentricity and subjected to two statistically independent components of excitation. Several practical problems require analysis for non-stationary excitations. For this important class of problems the original TELM requires linearization for a series of points in time to study the evolution of response statistics. This procedure turns out to be computationally onerous. As an approximate alternative, we propose the evolutionary TELM, ETELM. In particular, we adopt the concepts of the evolutionary process theory, to define an evolutionary TELS, ETELS. The ETELS approximately estimates the continuous time evolution of the design point by only one TELM analysis. This is the essence of its efficiency compared to the standard TELM analysis. Among response statistics of interest, the first-passage probability represents the most important one for this class of problems. This statistic is efficiently computed by using the Au-Beck important sampling algorithm, which requires knowledge of the evolving design points, in conjunction with the ETELS. The method is successfully tested for five types of excitation: (I) uniformly modulated white noise, (II) uniformly modulated broad-band excitation, (III) uniformly modulated narrow-band excitation, (IV) time- and frequency-modulated broad-band excitation, and (V) time- and frequency-modulated narrow-band excitation.