In his 1971 book Schnorr introduced a constructive measure that relates to Martin-L\"of's notion of constructive measure in much the same way as ``computable'' relates to ``computably enumerable''.  Schnorr carefully analyzed this notion, and he also characterized it in terms of computable martingales and orders of growth. Schnorr also considered the randomness notion corresponding to computable martingales. In this talk we discuss the differences between these randomness notions from various points of view.