1. Introduction

Semialgebraic and subanalytic sets capture ideas in several areas: In model theory, they express properties of quantifier elimination. In geometry and analysis, they provide a language for questions about the local behaviour of algebraic and analytic mappings. Lou van den Dries has suggested that the ominimal structure of the classes of semialgebraic or subanalytic sets makes precise Grothendieck's vision of a “tame topology”. (In his provocative Esquisse d'un programme, Grothendieck [1984] proposes an axiomatic development of a topology based on ideas of stratification in order to study, for example, singularities that arise in compactifications of moduli spaces.) These notes present another point of view (with intriguing possible relationships): we will describe a range of geometric classes of spaces between semialgebraic and subanalytic, that do not necessarily fit into the ominimal framework, but that are “tame” from algebraic or analytic perspectives. The questions we discuss are in directions pioneered by Whitney, Thorn, Lojasiewicz, Gabrielov and Hironaka. (We will not try to give a general survey of recent results in the area of semialgebraic and subanalytic sets.)