In this paper, we study quasicompact and Riesz composition endomorphisms of Lipschitz algebras of complex-valued bounded functions on metric spaces, not necessarily compact. We give some necessary and some sufficient conditions that a composition endomorphism of these algebras to be quasicompact or Riesz.We also establish an upper bound and a formula for the essential spectral radius of a composition endomorphism T of these algebras under some conditions which implies that T is quasicompact or Riesz. Finally, we get a relation for the set of eigenvalues and the spectrum of a quasicompact and Riesz endomorphism of these algebras.
