We present a summary of the history, present day activities and some possible future directions of molecular beam epitaxy. We confine our attention to growth-related phenomena and do not attempt to discuss the vast amount of work on device fabrication, although we acknowledge its huge contribution to the subject. We emphasize the extent to which basic studies are now being directed towards an understanding of growth processes at the atomistic level, treating as examples the homoepitaxial growth of GaAs(001) films and InAs-GaAs quantum dot formation. We also discuss recent work on silicon and on Group III-element nitrides. (C) 2004 Elsevier B.V. All rights reserved.