  A noncommutative solenoid is the C*-algebra $C^\ast(\Q_N^2,\sigma)$ where
$\Q_N$ is the group of the $N$-adic rationals twisted and $\sigma$ is a
multiplier of $\Q_N^2$. In this paper, we use techniques from noncommutative
topology to classify these C*-algebras up to *-isomorphism in terms of the
multipliers of $\Q_N^2$. We also establish a necessary and sufficient condition
for simplicity of noncommutative solenoids, compute their K-theory and show
that the $K_0$ groups of noncommutative solenoids are given by the extensions
of $\Z$ by $\Q_N$. We give a concrete description of non-simple noncommutative
solenoids as bundle of matrices over solenoid groups, and we show that
irrational noncommutative solenoids are real rank zero AT C*-algebras.
