Classical elimination theory culminated with the work of Francis Macaulay on multivariate resultants, as described in the chapter on Elimination theory in the first editions (1930) of Bartel van der Waerden's Moderne Algebra. After that, elimination theory was ignored by most algebraic geometers for almost thirty years, until the introduction of new methods for solving polynomial equations, such as Gröbner bases, which were needed for computer algebra.
Before the 20th century, different types of eliminants were introduced, including resultants, and various kinds of discriminants. In general, these eliminants are also invariant under various changes of variables, and are also fundamental in invariant theory.
All these concepts are effective, in the sense that their definitions include a method of computation. Around 1890, David Hilbert introduced non-effective methods, and this was seen as a revolution, which led most algebraic geometers of the first half of the 20th century to try to "eliminate elimination". Nevertheless Hilbert's Nullstellensatz, may be considered to belong to elimination theory, as it asserts that a system of polynomial equations does not have any solution if and only if one may eliminate all unknowns to obtain the constant equation 1 = 0.
Later, elimination theory was considered old-fashioned and removed from subsequent editions of Moderne Algebra. It was generally ignored until the introduction of computers, and more specifically of computer algebra, which again made relevant the design of efficient elimination algorithms, rather than merely existence and structural results. The main methods for this renewal of elimination theory are Gröbner bases and cylindrical algebraic decomposition, introduced around 1970.