A small-deformation perturbation analysis is developed to study the effect of surfactant on drop dynamics in viscous flows. The surfactant is assumed to be insoluble in the bulk-phase fluids; the viscosity ratio and surfactant elasticity parameters are arbitrary. Under small-deformation conditions, the drop dynamics are described by a system of ordinary differential equations; the governing equations are given explicitly for the case of axisymmetric and two-dimensional imposed flows. Analytical results accurate to third order in the flow-strength parameter (capillary number) are derived (i) for the stationary drop shape and surfactant distribution in simple shear and axisymmetric straining flows, and (ii) for the rheology of a dilute emulsion in shear flow which include a shear-thinning viscosity and non-zero normal stresses. For drops with clean interfaces, the small-deformation theory presented here improves the results of Barth's-Biesel & Acrivos (J. Fluid Mech., vol. 61, 1973, p. 1). Boundary integral simulations are used to test our theory and explore large-deformation conditions.