The notion of spectrum of an operator is one of the central concepts of operator theory. It is closely connected with the existence of a functional calculus which provides important information about the structure of Banach space operators. The situation for commuting n-tuples of Banach space operators is much more complicated. There are many possible definitions of joint spectra. However, the joint spectrum introduced by J. L. Taylor has a distinguished property—there exists a functional calculus for functions analytic on a neighbourhood of this spectrum, and the Taylor spectrum is in some sense minimal (so the Taylor functional calculus is the richest). The present chapter gives a survey of basic properties of the Taylor spectrum and Taylor functional calculus.