We describe recent developments in the theory of real recursive functions, as discovered by the speaker in collaboration with José Félix Costa and Jerzy Mycka. The class of real recursive functions is the smallest class of partial real-valued vector functions containing the functions 1, -1, 0, and the projections, and closed under composition, differential recursion and infinite limits. Restrictions of this inductive scheme have been previously shown, in this seminar, to correspond to the computable functions over R. We will study the most general and unrestricted form of this inductive scheme by: (1) giving a general introduction; (2) describing a few fundamental results, of which some are new; and, ultimately, (3) showing that real recursive functions have an exact correspondence with the analytical hierarchy of (second-order) predicates.