A Bose--Einstein condensate (BEC) is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near to absolute zero (0 K or -273.15 DegreesC). Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at which point quantum effects become apparent on a macroscopic scale. This book gathers and presents research in this field including a new approach to Spinor Bose-Einstein condensates, elliptic vortices in self-attractive Bose-Einstein condensates and matter wave dark solitions in optical superlatices, as well as the mathematical description of the effective behavior of one-dimensional Bose-Einstein condensates with defects.