This volume is a collection of written versions of the talks given at the Workshop on Computational Prospects of Infinity, held at the Institute for Mathematical Sciences from 18 June to 15 August 2005. It consists of contributions from many of the leading experts in recursion theory (computability theory) and set theory. Topics covered include the structure theory of various notions of degrees of unsolvability, algorithmic randomness, reverse mathematics, forcing, large cardinals and inner model theory, and many others. Contents: Prompt Simplicity, Array Computability and Cupping (R Downey et al.); A Simpler Short Extenders Forcing Gap 3 (M Gitik); The Strength of Some Combinatorial Principles Related to Ramsey's Theorem for Pairs (D R Hirschfeldt et al.); Absoluteness for Universally Baire Sets and the Uncountable II (I Farah et al.); Modaic Definability of Ordinals (I Neeman); Eliminating Concepts (A Nies); Rigidity and Biinterpretability in the Hyperdegrees (R A Shore); Some Fundamentals Issues Concerning Degrees or Unsolvability (S G Simpson); A tt Version of the Posner Robinson Theorem (W H Woodin); and other papers.