  In the last three decades, several constructions of quantum error-correcting
codes were presented in the literature. Among these codes, there are the
asymmetric ones, i.e., quantum codes whose $Z$-distance $d_z$ is different from
its $X$-distance $d_x$. The topological quantum codes form an important class
of quantum codes, where the toric code, introduced by Kitaev, was the first
family of this type. After Kitaev's toric code, several authors focused
attention on investigating its structure and the constructions of new families
of topological quantum codes over Euclidean and hyperbolic surfaces. As a
consequence of establishing the existence and the construction of asymmetric
topological quantum codes in Theorem \ref{main}, the main result of this paper,
we introduce the class of hyperbolic asymmetric codes. Hence, families of
Euclidean and hyperbolic asymmetric topological quantum codes are presented. An
analysis regarding the asymptotic behavior of their distances $d_x$ and $d_z$
and encoding rates $k/n$ versus the compact orientable surface's genus is
provided due to the significant difference between the asymmetric distances
$d_x$ and $d_z$ when compared with the corresponding parameters of topological
codes generated by other tessellations. This inherent unequal error-protection
is associated with the nontrivial homological cycle of the $\{p,q\}$
tessellation and its dual, which may be appropriately explored depending on the
application, where $p\neq q$ and $(p-2)(q-2)\ge 4$. Three families of codes
derived from the $\{7,3\}$, $\{5,4\}$, and $\{10,5\}$ tessellations are
highlighted.
