The field of rational numbers is not such a number field. That is, there does not exist a smooth projective morphism $X\to\text{Spec } \mathbf{Z}$ such that the generic fibre is a curve of genus $\geq 1$.
Which number fields allow (or do not allow) the existence of such curves?
For any number field $K/\mathbf{Q}$ of degree $>1$, does there exist a smooth projective geometrically connected curve $X$ over $K$ with good reduction over $K$?