This is a survey of constructive and computable measure theory with an emphasis on the close connections with algorithmic randomness. We give a brief history of constructive measure theory from Brouwer to the present, emphasizing how Schnorr randomness is the randomness notion implicit in the work of Brouwer, Bishop, Demuth, and others. We survey a number of recent results showing that classical almost everywhere convergence theorems can be used to characterize many of the common randomness notions including Schnorr randomness, computable randomness, and Martin-Löf randomness. Last, we go into more detail about computable measure theory, showing how all the major approaches are basically equivalent (even though the definitions can vary greatly).