  We establish a combinatorial realization of continued fractions as quotients
of cardinalities of sets. These sets are sets of perfect matchings of certain
graphs, the snake graphs, that appear naturally in the theory of cluster
algebras. To a continued fraction $[a_1,a_2,\ldots,a_n]$, we associate a snake
graph $\mathcal{G}[a_1,a_2,\ldots,a_n]$ such that the continued fraction is the
quotient of the number of perfect matchings of
$\mathcal{G}[a_1,a_2,\ldots,a_n]$ and $\mathcal{G}[a_2,\ldots,a_n]$. We also
show that snake graphs are in bijection with continued fractions.
  We then apply this connection between cluster algebras and continued
fractions in two directions. First, we use results from snake graph calculus to
obtain new identities for the continuants of continued fractions. Then, we
apply the machinery of continued fractions to cluster algebras and obtain
explicit direct formulas for quotients of elements of the cluster algebra as
continued fractions of Laurent polynomials in the initial variables. Building
on this formula, and using classical methods for infinite periodic continued
fractions, we also study the asymptotic behavior of quotients of elements of
the cluster algebra.
