Analysis of the trajectories of small particles at high spatial and temporal resolution using video enhanced contrast microscopy provides a powerful approach to characterizing the mechanisms of particle motion in living cells and in other systems. We present here the theoretical basis for the analysis of these trajectories for particles undergoing random diffusion and/or systematic transport at uniform velocity in two-dimensional systems. The single particle tracking method, based on observations of the trajectories of individual particles, is compared with methods that characterize the motions of a large collection of particles such as fluorescence photobleaching recovery. Determination of diffusion coefficients or transport velocities either from correlation of positions or of velocities of the particles is discussed. A result of practical importance is an analysis of the dependence of the expected statistical uncertainty of these determinations on the number of position measurements. This provides a way of judging the accuracy of the diffusion coefficients and transport velocities obtained using this approach.