  We show that CSP is fixed-parameter tractable when parameterized by the
treewidth of a backdoor into any tractable CSP problem over a finite constraint
language. This result combines the two prominent approaches for achieving
tractability for CSP: (i) by structural restrictions on the interaction between
the variables and the constraints and (ii) by language restrictions on the
relations that can be used inside the constraints. Apart from defining the
notion of backdoor-treewidth and showing how backdoors of small treewidth can
be used to efficiently solve CSP, our main technical contribution is a
fixed-parameter algorithm that finds a backdoor of small treewidth.
