  A graph $G$ is $\textit{universal}$ for a (finite) family $\mathcal{H}$ of
graphs if every $H \in \mathcal{H}$ is a subgraph of $G$. For a given family
$\mathcal{H}$, the goal is to determine the smallest number of edges an
$\mathcal{H}$-universal graph can have. With the aim of unifying a number of
recent results, we consider a family of graphs with bounded density. In
particular, we construct a graph with $O_d\left( n^{2 - 1/(\lceil d \rceil +
1)} \right)$ edges which contains every $n$-vertex graph with density at most
$d \in \mathbb{Q}$ ($d \ge 1$), which is close to a lower bound $\Omega(n^{2 -
1/d - o(1)})$ obtained by counting lifts of a carefully chosen (small) graph.
When restricting the maximum degree of such graphs to be constant, we obtain a
near-optimal universality. If we further assume $d \in \mathbb{N}$, we get an
asymptotically optimal construction.
