  We proceed to study Yang-Baxter deformations of the AdS$_5\times$S$^5$
superstring with the classical Yang-Baxter equation. We make a general argument
on the supercoset construction and present the master formula to describe the
dilaton in terms of classical $r$-matrices. The supercoset construction is
explicitly performed for some classical $r$-matrices and the full backgrounds
including the Ramond-Ramond (R-R) sector and dilaton are derived. Within the
class of abelian $r$-matrices, the perfect agreement is shown for well-known
examples including gravity duals of non-commutative gauge theories,
$\gamma$-deformations of S$^5$ and Schr\"odinger spacetimes. It would be
remarkable that the supercoset construction works well, even if the resulting
backgrounds are not maximally supersymmetric. In particular, three-parameter
$\gamma$-deformations of S$^5$ and Schr\"odinger spacetimes do not preserve any
supersymmetries. As for non-abelian $r$-matrices, we will focus upon a specific
example. The resulting background does not satisfy the equation of motion of
the Neveu-Schwarz-Neveu-Schwarz (NS-NS) two-form because the R-R three-form is
not closed.
