  We develop a general theory which enables the computation of the Picard group
of a symmetric monoidal triangulated category, equipped with a weight
structure, in terms of the Picard group of the associated heart. As an
application, we compute the Picard group of several categories of motivic
nature - mixed Artin motives, mixed Artin-Tate motives, motivic spectra,
noncommutative mixed Artin motives, noncommutative mixed motives of central
simple algebras, noncommutative mixed motives of separable algebras - as well
as the Picard group of the derived categories of symmetric ring spectra.
