  This paper defines several algebras associated to an oriented surface $S$
with a finite set of marked points on the boundary. The first is the skein
algebra $Sk_q(S)$, which is spanned by links in the surface which are allowed
to have endpoints at the marked points, modulo several locally defined
relations. The product is given by superposition of links. A basis of this
algebra is given, as well as several algebraic results.
  When $S$ is triangulable, the quantum cluster algebra $A_q(S)$ and quantum
upper cluster algebra U_q(S) can be defined. These are algebras coming from the
triangulations of S and the elementary moves between them.
  Natural inclusions $A_q(S)$ into $Sk_q^o(S)$ into $U_q(S)$ are shown, where
$Sk_q^o(S)$ is a certain Ore localization of $Sk_q(S)$. When $S$ has at least
two marked points in each component, these inclusions are strengthened to
equality, exhibiting a quantum cluster structure on $Sk_q^o(S)$.
  The method for proving these equalities has potential to show $A_q=U_q$ for
other classes of cluster algebras. As a demonstration of this fact, a new proof
is given that $A_q=U_q$ for acyclic cluster algebras
