  We provide methods to construct explicit examples of $K3$ surfaces. This
leads to unirational constructions of Noether--Lefschetz divisors inside the
moduli space of $K3$ surfaces of genus $g$. We implement Mukai's unirationality
construction of the moduli spaces of $K3$ surfaces of genus $g\in\{6,\dots, 10,
12\}$, and we also present a new constructive proof of the unirationality of
the moduli space of $K3$ surfaces of genus $11$. Furthermore, we show the
existence of three unirational hypersurfaces in any moduli space of $K3$
surfaces of genus $g$.
