A historical perspective on and substantive review of the concept of invariance are provided. Progress made toward solving measurement problems related to invariance is also assessed. Two major classes of invariant measurement are described: (1) sample-invariant item calibration; and (2) item-invariant measurement of individuals. The work of S. S. Stevens is used to help clarify the concept of invariance. The importance of invariance as a key measurement concept is then illustrated via the measurement theories of E. L. Thorndike, L. L. Thurstone, and G. Rasch. The study methodology uses quotations and original figures to illustrate how these researchers addressed measurement problems related to invariance. A comparison and discussion of these three researchers' theories of measurement are presented in terms of their contributions to the solution of problems related to the concept of invariance. Rasch's research is seen as the means by which the issues raised by the other two researchers were resolved. A case is made for viewing invariance as a fundamental aspect of measurement in the behavioral sciences. Invariance appears to be essential in order to realize the advantages of objective measurement. A 58-item list of references, one table, and five figures are included. (TJH)