  We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which
is the union of components whose general point corresponds to a smooth
irreducible and non-degenerate curve of degree $d$ and genus $g$ in
$\mathbb{P}^r$. In this article, we study $\mathcal{H}_{15,g,5}$ for every
possible genus $g$ and determine their irreducibility. We also study the
birationality of the moduli map up to projective equivalence and several key
properties such as gonality of a general element as well as characterizing
smooth elements of each component.
