Recently the concept of regression depth has been introduced [J. Am. Stat. Assoc. 94 (1999) 388]. The deepest regression (DR) is a method for linear regression which is defined as the fit with the best depth relative to the data. In this paper we explain the properties of the DR and give some applications of DR in analytical chemistry which involve regression through the origin, polynomial regression, the Michaelis–Menten model, and censored responses.