This self-contained reference/text presents a thorough account of the theory of real function algebras. Employing the intrinsic approach, avoiding the complexification technique, and generalizing the theory of complex function algebras, this single-source volume includes: an introduction to real Banach algebras, various generalizations of the Stone-Weierstrass theorem, Gleason parts, Choquet and Shilov boundaries, isometries of real function algebras, extensive references, and a detailed bibliography., Real Function Algebras offers results of independent interest such as: topological conditions for the commutativity of a real or complex Banach algebra, Ransford's short elementary proof of the Bishop-Stone-Weierstrass theorem, the implication of the analyticity or antianalyticity of f from the harmonicity of Re f, Re f(2), Re f(3), and Re f(4), and the positivity of the real part of a linear functional on a subspace of C(X)., With over 600 display equations, this reference is for mathematical analysts, pure, applied, and industrial mathematicians, and theoretical physicists, and a text for courses in Banach algebras and function algebras.