 Kleene introduced what is now known as the analytical hierarchy in a series of papers (1955a,b,c) which built directly on his introduction of the arithmetical hierarchy in (1943). His proximal motivation was to provide a definability-theoretic characterization of the so-called hyperarithmetical sets—i.e., those which are computable from transfinite iterates of the Turing jump through the constructive ordinals. On the other hand, Mostowski (1947) had already noticed similarities between the arithmetical hierarchy of sets of natural numbers and results about hierarchies of point sets studied in descriptive set theory—i.e., sets of elements of Polish spaces (complete, separable metrizable spaces such as the real numbers, Cantor space, or Baire space)—which have their origins in the work of Borel, Lebesgue, Lusin, and Suslin in the early twentieth century. Beginning in his PhD thesis under Kleene, Addison (1954) refined Mostowski’s comparisons by showing that Kleene’s initial work could also be used to provide effective versions of several classical results in this tradition. We present here the fundamental definitions regarding the analytical hierarchy together with some of some results illustrating how it is connected it to these other developments.