I will describe the construction of a new database of modular curves that is to become part of the L-functions and Modular Forms Database (LMFDB). This database is organized around open subgroups $H$ of the profinite group $\mathrm{GL}(2,\widehat{\mathbb{Z}})$ that determine modular curves $X_H$ that parameterizes elliptic curves with "level-$H$ structure" given by the Galois-action on torsion points. This notably includes classical modular curves such as $X_0(N)$ and $X_1(N)$ that are traditionally defined as quotients of the upper half-plane by congruence subgroups in $\mathrm{SL}(2,\mathbb{Z})$ but also includes many more exotic examples.