  This article provides an expository account of the celebrated duality theorem
of Bavard and three its strengthenings. The Bavard duality theorem connects scl
(stable commutator length) and quasimorphisms on a group. Calegari extended the
framework from a group element to a chain on the group, and established the
generalized Bavard duality. Kawasaki, Kimura, Matsushita and Mimura studied the
setting of a pair of a group and its normal subgroup, and obtained the mixed
Bavard duality. The first half of the present article is devoted to an
introduction to these three Bavard dualities. In the latter half, we present a
new strengthening, the generalized mixed Bavard duality, and provide a
self-contained proof of it. This third strengthening recovers all of the Bavard
dualities treated in the first half; thus, we supply complete proofs of these
four Bavard dualities in a unified manner. In addition, we state several
results on the space $\mathrm{W}(G,N)$ of non-extendable quasimorphisms, which
is related to the comparison problem between scl and mixed scl via the mixed
Bavard duality.
