  Classically, congruence subgroups of the modular group, which can be
described by congruence relations, play important roles in group theory and
modular forms. In reality, the majority of finite index subgroups of the
modular group are noncongruence. These groups as well as their modular forms
are central players of this survey article. Differences between congruence and
noncongruence subgroups and modular forms will be discussed. We will mainly
focus on three interesting aspects of modular forms for noncongruence
subgroups: the unbounded denominator property, modularity of the Galois
representation arising from noncongruence cuspforms, and Atkin and
Swinnerton-Dyer congruences.
