Frolik did the early work with almost realcompactness in the early 60's, and Dykes the work with a-realcompactness towards the end of that decade. In a 1980 paper Archangel'skii produced a generalization of a-realcompact spaces he called pure spaces. Pure spaces were further generalized by Sekai in 1986 and given the name neat spaces. The definitions of both pure and neat spaces are well beyond the scope of this little survey. In a 1970 paper Dykes defined c-realcompactness essentially by weakening the continuity criterion on f to normal lower semicontinuity.
Dykes demonstrates in that paper that almost realcompact spaces are c-realcompact.
Rings of continuous functions were used successfully by Mandelker in 1971 and by Johnson and Mandelker in 1973 to explore still other ways to generalize realcompactness. As promised, each of these turn out to generalize realcompactness, and Johnson and Mandelker showed that all these concepts are distinct. Yet another trail of generalizations can be considered as descending from our fourth characterization of realcompactness. Shirota demonstrated that realcompact spaces are completely uniformizable in 1952. In fact, if one assumes that there are no measurable cardinals, the notions are equivalent.