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Mathematics Subject Classification (1991): l lGxx, 14Axx, 14GxxWhen J. Tate introduced rigid spaces in IT ], he was influenced by Grothendieck's idea of associating a generic fibre to a formal scheme satisfying certain finiteness conditions. After Tate's notes IT] had been communicated, considerable efforts were made, for example by the school of Grauert and Remmert, to develop rigid geometry in terms of analytic methods, in analogy to complex analysis. On the other hand, it was Raynaud [R] who suggested to view rigid spaces entirely within the framework of formal schemes. Both approaches have their advantages. Most notably, the approach through formal geometry allows the application of powerful methods from algebraic geometry, thus leading to rigorous solutions of various problems which, from a strictly analytic point of view, can only be dealt with in an ad-hoc-manner.Apart from the colloquium talk JR], the approach to rigid geometry via formal schemes is not well-documented. It is our intention to elaborate the ideas of Raynaud in order to pave the way for accessing some interesting applications. The present paper is of introductory nature. Its purpose is to motivate and explain Raynaud's definition of the category of rigid spaces as a localization of the appropriate category of formal schemes by admissible formal blowing-ups. This way it is possible to define rigid spaces over complete noetherian rings, not just over complete valuation rings.In Sects. 1 and 2 the basic results and constructions concerning admissible formal schemes are explained, among them the technique of admissible formal blowing-up. Then, in Sect. 3, we define rig-points of admissible formal schemes, which later on are interpreted as points of rigid spaces. In some sense, they provide the link between an admissible formal scheme and its associated rigid space. After having gathered these technical prerequisites, we show in Sect. 4 how to interpret classical rigid spaces in terms of formal schemes, in the way it was indicated by Raynaud in [R]. Finally, in Sect. 5 the approach is extended to Raynaud's relative rigid spaces over a global noetherian base.One can start now and generalize classical rigid geometry to the relative case. The point of departure for any activity in this field is the basic result of Raynaud

9 Springer-Verlag 1985 Proposition 1.2. Let X u be distinguished and assume that Xu is geometrically reduced. Then, for any complete field K extending k, the formal analytic variety X~ is distinguished. Its reduction X~ is canonically isomorphic to the I(-extension of the reduction )(u of X u. Proof. Use [B 1, Satz 6.4]. Proposition 1.3. Assume that X is quasi-compact (i.e., that X admits a finite admissible open affinoid covering), and that X is geometrically reduced. Then there exists a finite separable extension k' of k such that X k' is distinguished and has geometrically reduced reduction )~k'. Proof. Use [B2, Lemma 2.7].Now let k-be the completion of the algebraic closure of k. We say that a formal covering ~B of X~is defined over k if there exists a formal covering Il of X such that = 1/~. Likewise we say that X~ is defined over k.Proposition _1.4. Assume that X is quasi-compact, and let ~ be a formal covering of X k. Then X k is defined over a finite separable extension k' of k.

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