A fundamental understanding of oscillations in air of surfactant-laden drops is important in applications ranging from the measurement of interfacial properties to atomization and spraying of liquids from nozzles. Here the small- and large-amplitude oscillations of a drop of an incompressible Newtonian liquid that is covered with a monolayer of an insoluble surfactant are analyzed by solving the Navier-Stokes system governing liquid motion in the drop and the convection-diffusion equation governing surfactant transport along the liquid-gas interface. The former are determined by means of a normal mode analysis and the latter are analyzed by a computational technique based on a method-of-lines where the finite element method (FEM) is used for spatial discretization and adaptive finite differences are used for time integration. The FEM algorithm used employs an elliptic mesh generation technique, which enables study of extremely large-amplitude deformations, e.g. initial drop aspect ratios of O(10) or larger. Oscillation frequencies and damping rates are determined as functions of Reynolds number, dimensionless amplitude of the initial deformation, dimensionless initial surfactant concentration, surfactant activity, and Peclet number. The presence of surfactant, on the one hand, lowers surface tension and thereby increases the period of oscillation. Its presence, on the other hand, generates surface tension gradients and thus enhances damping of oscillations. The computations are used to determine how the frequency of oscillation of surfactant-covered drops falls as initial disturbance amplitude rises. The role of vorticity on the dynamics is also examined. Insights gained from it are used to rationalize the variation of the damping rate of oscillations as a function of the initial surfactant loading. The analytical and computational results are also employed to shed light on recent experimental measurements.