The author reviews some of the developments connected with testing statistical hypotheses together with several controverses on this phase of statistics. Ideas concerning the following are presented: definition of a statistical hypothesis, 2 views of probability, meaning of testing hypotheses, definitions of the 2 kinds of errors arising in making test, the power function, the power function of a most powerful test, regions and similar regions, tests which are unbiased, tests concerning randonization testing by use of x 2, tables of power functions, the power function of x 2, etc. Assumptions underlying the test are given, together with difficulties arising in applying them. This synopsis of the developments involved in testing statistical hypotheses is well written, contains many recent discoveries, and shows the rapid progress made in statistical analyses. The appendix discusses the work of R. A. Fisher and Jeffrey.