What I remember is that first Laver solved the Borel conjecture by an countable support iteration which added a real at every stage (1976). That such an iteration can be of any use, was very surprising at the time. Then Baumgartner introduced the very general notion of property A, which included most (all?) standard forcings known then which added reals and showed that countable support iteration of them behaves nicely (1978). Then came Shelah, who gave the proper definition (1980). This was again a surprising thing, as Baumgartner's definition was combinatorial (containing combinatorial properties of the poset, eh, almost) while Shelah simply required that P should preserve all stationary subsets of all sets of the form $[A]^{\aleph_0}$, that is, a semantic definition. Notice that Shelah's new theory gave new and elegant proofs to old theorems, as Baumgartner's consistency of that any two $\aleph_1$-dense sets of reals are isomorphic or Mitchell's consistency of the tree property of $\aleph_2$.