This paper is essentially the author's Gödel Lecture at the ASL Logic
Colloquium '09 in Sofia extended and supplemented by material from some other
papers. After a brief description of traditional reverse mathematics, a
computational approach to is presented. There are then discussions of some
interactions between reverse mathematics and the major branches of
mathematical logic in terms of the techniques they supply as well as theorems
for analysis. The emphasis here is on ones that lie outside the usual
main systems of reverse mathematics. While retaining the usual base theory and
working still within second order arithmetic, theorems are described that
range from those far below the usual systems to ones far above.