Adaptive array systems have undergone enormous growth and progress in the past two decades. Because this multidiscipline technology area is both complex and mathematics intensive, it often appears alien (and perhaps impractical) to the nonspecialist. To help bridge those gaps, this tutorial paper has been and limited to just three principal topic areas: sidelobe cancellers, fully adaptive arrays, and superresolution techniques. In each case, background material is included for perspective and clarity, and carefully selected references are provided. Recent developments are discussed briefly where appropriate.>