  We study the uniform random graph $\mathsf{C}_n$ with $n$ vertices drawn from
a subcritical class of connected graphs. Our main result is that the rescaled
graph $\mathsf{C}_n / \sqrt{n}$ converges to the Brownian Continuum Random Tree
$\mathcal{T}_{\mathsf{e}}$ multiplied by a constant scaling factor that depends
on the class under consideration. In addition, we provide subgaussian tail
bounds for the diameter $\text{D}(\mathsf{C}_n)$ and height
$\text{H}(\mathsf{C}_n^\bullet)$ of the rooted random graph
$\mathsf{C}_n^\bullet$. We give analytic expressions for the scaling factor of
several classes, including for example the prominent class of outerplanar
graphs. Our methods also enable us to study first passage percolation on
$\mathsf{C}_n$, where we show the convergence to $\mathcal{T}_{\mathsf{e}}$
under an appropriate rescaling.
