Readers acquainted with the differential calculus and the concept of a polynomial can effortlessly acquire an excellent perspective about invariants for differential equations by using a system of computer algebra to interact with the examples in Chapters 16 and 17. During the years 1879--1889, there were differential equations for which mathematicians found particular combinations of the coefficients that possessed an invariant character under unrestricted transformations. In fact, specific examples of basic relative invariants as the most interesting kind were discovered by E. Laguerre in 1879, G.-H. Halphen in 1880-1884, A. R. Forsyth in 1888, and P. Appell in 1989. However, there was little progress after 1889 about the principal problems until the subject was completely redeveloped during 1989--2014. A thorough explanation for this remarkable situation is given in Chapters 15 and 18. All of the basic relative invariants are now explicitly known for numerous types of differential equations and the main problems have now been solved. This monograph provides details about these developments and gives numerous illustrations to show how the relative invariants of a given weight are expressible in terms of the basic relative invariants.