Twenty years ago morphological analysis of natural language was a challenge to computational linguists. Simple cut-and-paste programs could be and were written to analyze strings in particular languages, but there was no general language-independent method available. Furthermore, cut-and-paste programs for analysis were not reversible, they could not be used to generate words. Generative phonologists of that time described morphological alternations by means of ordered rewrite rules, but it was not understood how such rules could be used for analysis. This was the situation in the spring of 1981 when Kimmo Koskenniemi came to a conference on parsing that Lauri Karttunen had organized at the University of Texas at Austin. Also at the same conference were two Xerox researchers from Palo Alto, Ronald M. Kaplan and Martin Kay. The four Ks discovered that all of them were interested and had been working on the problem of morphological analysis. Koskenniemi went on to Palo Alto to visit Kay and Kaplan at PARC. This was the beginning of Two-Level Morphology, the first general model in the history of computational linguistics for the analysis and generation of morphologically complex languages. The language-specific components, the lexicon and the rules, were combined with a runtime engine applicable to all languages. In this article we trace the development of the finite-state technology that Two-Level Morphology is based on. 1 The Origins Traditional phonological grammars, formalized in the 1960s by Noam Chomsky and Morris Halle (Chomsky and Halle, 1968) , consisted of an ordered sequence of rewrite rules that converted abstract phonological representations into surface forms through a series of intermediate representations. Such rules have the general form x-> y / z w where x, y, z, and w can be arbitrarily complex strings or feature-matrices. In mathematical linguistics (Partee et al., 1993), such rules are called CONTEXT-SENSITIVE REWRITE RULES, and they are more powerful than regular expressions or context-free rewrite rules.