The interest in membranes and higher dimensional extended geometrical objects was inspired by the great successes of the string and superstring, first in 1968-73 as a theory of hadrons and then since 1984 as a “theory of everything” — a unified theory of all interactions, including quantum gravity. In particular, membranes, “supermembranes” and “spinning membranes” have been studied since 1985; an unexpected connection with supergravity opened in 1988 some new prospects for a quantized theory. General theorems about higher dimensional “p-branes” or “extendons” were also derived. The subject is very promising, whether as a more comprehensive fundamental unification or as a description of composite hadrons.Where there are now probably some fifty texts of all kinds dealing with strings, this is the first treatise on membranes and higher dimensional extendons.