This book examines ultrametric Banach algebras in general. It begins with algebras of continuous functions and looks for maximal ideals and prime ideals, in connection with ultrafilters on,nthe set of definition. The multiplicative spectrum has shownto be indispensable in ultrametric analysis and is dercribed in the general context and then in various cases of Banach algebras. Applications are made to various kind of functions: uniformly continuous functions, Lipschitz functions, strictly differentiable functions, defined in a. metric space. Analytic. elements in an algebraically closed complete field (due to M. Krasner). are recalled with most of their properties linked to T-filters and applications. to their Banach algebras and to the holomorphic functional calculus, with applications to spectral properties. The multiiplicative semi-norms. of Krasner algebras are characterized by circular filters. with a. metric and an order that are examined. All the main properties of affinoid algebras are recalled, including _Krasner-Tate algebras and the existence of idempotents associated to connected components of the mutiplicative spectrum is described.