  We study the modular symmetry in heterotic string theory on Calabi-Yau
threefolds. In particular, we examine whether moduli-dependent holomorphic
Yukawa couplings are described by modular forms in the context of heterotic
string theory with standard embedding. We find that $SL(2,\mathbb{Z})$ modular
symmetry emerges in asymptotic regions of the Calabi-Yau moduli space. The
instanton-corrected holomorphic Yukawa couplings are then given by modular
forms under $SL(2,\mathbb{Z})$ or its congruence subgroups such as
$\Gamma_0(3)$ and $\Gamma_0(4)$. In addition to the modular symmetry, it turns
out that another coupling selection rule controls the structure of holomorphic
Yukawa couplings. Furthermore, the coexistence of both the positive and
negative modular weights for matter fields leads to a hierarchical structure of
matter field K\"ahler metric. Thus, these holomorphic modular forms and the
matter field K\"ahler metric play an important role in realizing a hierarchical
structure of physical Yukawa couplings.
