  In this review article, we present a unified approach to solving discrete,
integrable, possibly non-commutative, dynamical systems, including the $Q$- and
$T$-systems based on $A_r$. The initial data of the systems are seen as cluster
variables in a suitable cluster algebra, and may evolve by local mutations. We
show that the solutions are always expressed as Laurent polynomials of the
initial data with non-negative integer coefficients. This is done by
reformulating the mutations of initial data as local rearrangements of
continued fractions generating some particular solutions, that preserve
manifest positivity. We also show how these techniques apply as well to
non-commutative settings.
