We survey recent results on the connection between Nonstandard Analysis (NSA) and computability theory. We present a number of negative and positive results and how they translate back and forth between NSA and computability theory. Of particular interest is how a negative result regarding the axioms of NSA (namely 'Transfer =/=> Standardization') gives rise to two fundamentally different paths from second-order to higher-order mathematics (and the associated notions of computability).