Several alternative methods for directly integrating the governing equations of structural dynamics are reviewed. First, characteristics of the matrix equations, such as the spread in structural eigenvalues, or stiffness, the bandwidth and sparseness, and the frequency spectrum of the forcing function, are examined. Then, criteria that can be used to select a direct integration algorithm, such as artificial damping and periodicity error, are analyzed. Emphasis is given to results obtained for the Houbolt, Newmark, and Wilson operators, and their comparison to a class of stiffly stable operators. Correspondence of the Newmark method to the trapezoidal rule for γ = 1/2, β = 1/4 is shown. Recent application of these operators to nonlinear problems is covered.