In this paper I argue that cross-linguistic grammatical comparison cannot be based on grammatical categories, because these are language-specific. Instead, typology must be (and usually is) based on a special set of comparative concepts that are specifically created by typologists for the purposes of comparison. Descriptive formal categories cannot be equated across languages because the criteria for category-assignment are different from language to language. This old structuralist insight (called categorial particularism) has recently been emphasized again by several linguists, but the idea that typologists need to identify "crosslinguistic categories" before they can compare languages is still widespread. Instead, what they have to do (and normally do in practice) is to create comparative concepts that help them to identify comparable phenomena across languages and to formulate cross-linguistic generalizations. Comparative concepts have to be universally applicable, so they can only be based on other universally applicable concepts: conceptual-semantic concepts, formal concepts, general concepts, and other comparative concepts. If, by contrast, one espouses categorial universalism and assumes crosslinguistic categories, as many generative linguists do, typology works by equating comparable categories in different languages, which are said to "instantiate" a cross-linguistic category. But in typological practice, all that is required is that a language-specific category matches a comparative concept. For example, the Russian Dative, the Turkish Dative and the Finnish Allative all match the comparative concept 'dative case', but they are very different distributionally and semantically and therefore cannot be equated and cannot instantiate a cross-linguistic category 'dative'. Comparative concepts are not always purely semantically-based concepts, but outside of phonology they usually contain some semantic components. If one is not confident about the universality of meanings, one can substitute extralinguistic contexts for universal meanings. The view that descriptive categories are different across languages and different from comparative concepts leads to terminological problems, which are also discussed here. Finally, I observe that the adoption of categorial universalism has actually impeded, not facilitated, cross-linguistic research.