  The $SU_3$-skein algebra of a surface $F$ is spanned by isotopy classes of
certain framed graphs in $F\times I$ called $3$-webs subject to the skein
relations encapsulating relations between $U_q(sl(3))$-representations. These
skein algebras are quantizations of the $SL(3)$-character varieties of
surfaces. It is expected that their theory parallels that of the Kauffman
bracket skein algebras. We make the first step towards developing that theory
by proving that the reduced $SU_3$-skein algebra of any surface of finite type
is finitely generated.
  We achieve that result by developing a theory of canonical forms of webs in
surfaces. Specifically, we show that for any ideal triangulation of $F$ every
reduced $3$-web can be uniquely decomposed into unions of pyramid formations of
hexagons and disjoint arcs in the faces of the triangulation with possible
additional "crossbars" connecting their edges along the ideal triangulation. We
show that such canonical position is unique up to "crossbar moves". That leads
us to an associated system of coordinates for webs in triangulated surfaces
(counting intersections of the web with the edges of the triangulation and
their rotation numbers inside of the faces of the triangulation) which
determine a reduced web uniquely.
  Finally, we relate our skein algebras to $\cal A$-varieties of Fock-Goncharov
and to $\text{Loc}_{SL(3)}$-varieties of Goncharov-Shen. We believe that our
coordinate system for webs is a manifestation of a (quantum) mirror symmetry
conjectured by Goncharov-Shen.
