In this paper we introduce a new approach to determinant functors
which allows us to extend Deligne's determinant functors for exact categories
to Waldhausen categories, (strongly) triangulated categories, and derivators.
We construct universal determinant functors in all cases by original methods
which are interesting even for the known cases. Moreover, we show that the
target of each universal determinant functor computes the corresponding K-
theory in dimensions 0 and 1. As applications, we answer open questions by
Maltsiniotis and Neeman on the K-theory of (strongly) triangulated categories
and a question of Grothendieck to Knudsen on determinant functors. We also
prove additivity and localization theorems for low-dimensional K-theory and
obtain generators and (some) relations for various K1-groups.