A Bose-Einstein condensate (BEC) of a dilute gas is a unique quantum system with many atoms occupying the same quantum state. We are interested in all aspects of BECs including their formation, their collective properties, their coherence properties, and their use as sources for an atom laser. We have observed four-wave mixing of matter waves, the first demonstration of nonlinear atom optics, and the propagation of a dark soliton in the BEC, created with phase imprinting. Our current research topics include the investigation of the physics of BECs confined in periodic potentials (optical lattices) and the use of this system for investigating theoretical models in condensed matter; the study of the coherence and superfluid properties of the BEC, especially in ring-shaped traps; and the study of the phase transition and coherence properties of a BEC due to spin interactions. We are also studying spin dynamics in sodium spinor condensates and spontaneous 4-wave mixing processes in spin-changing collisions.