  Given a finite index subgroup $\Gamma$ of ${\rm{PSL}}_2(\Bbb{Z})$, we
investigate Belyi functions on the corresponding modular curve $X(\Gamma)$ by
introducing two methods for constructing such functions. Numerous examples have
been worked out completely and as an application, we have derived modular
equations for $\Gamma_0(2),\Gamma_0(3)$ and several special values of the
$j$-function by a new method based on the theory of Belyi functions and dessin
d'enfants.
