  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 theorems for low-dimensional $K$-theory and obtain generators and
(some) relations for various $K_{1}$-groups.
