The main thesis of the paper is that in the case of modern statistics, the differences between the various concepts of models were the key to its formative controversies. The mathematical theory of statistical inference was mainly developed by Ronald A. Fisher, Jerzy Neyman, and Egon S. Pearson. Fisher on the one side and Neyman–Pearson on the other were involved often in a polemic controversy. The common view is that Neyman and Pearson made Fisher's account more stringent mathematically. It is argued, however, that there is a profound theoretical basis for the controversy: both sides held conflicting views about the role of mathematical modelling. At the end, the influential programme of Exploratory Data Analysis is considered to be advocating another, more instrumental conception of models. Introduction Models in statistics—‘of what population is this a random sample?’ The fundamental lemma Controversy about models Exploratory data analysis as a model-critical approach.