Using extensive molecular dynamics simulations of an equilibrium; glass-forming Lennard-Jones mixture; we characterize in detail the local atomic motions. We show that spatial correlations exist among particles undergoing extremely large (“mobile”) or extremely small (“immobile”) displacements over a suitably chosen time interval. The immobile particles form the cores of relatively compact clusters; while the mobile particles move cooperatively and form quasi-one-dimensional; stringlike clusters. The strength and length scale of the correlations between mobile particles are found to grow strongly with decreasing temperature; and the mean cluster size appears to diverge near the mode-coupling critical temperature. We show that these correlations in the particle displacements are related to equilibrium fluctuations in the local potential energy and local composition.