This study is concerned with the underlying statistical theory required for tests of significance in sequential multiple linear regression analysis and with difficulties that can arise in developing valid tests. The relationship of the general linear subhypothesis theory to the Forward and Backward Selection Procedures is developed. The biases in estimation and the tests of significance are developed for the Forward Selection, Backward Elimination and Stagewise Regression procedures. The All Possible Regressions Procedure is described and a method for reducing the computational effort is outlined. Finally, the Stepwise Regression Procedure is considered and a Monte Carlo study of this procedure for the two variable case is discussed.