  A graph $U$ is universal for a graph class $\mathcal{C}\ni U$, if every $G\in
\mathcal{C}$ is a minor of $U$. We prove the existence or absence of universal
graphs in several natural graph classes, including graphs component-wise
embeddable into a surface, and graphs forbidding $K_5$, or $K_{3,3}$, or
$K_\infty$ as a minor. We prove the existence of uncountably many minor-closed
classes of countable graphs that (do and) do not have a universal element.
  Some of our results and questions may be of interest to the finite graph
theorist. In particular, one of our side-results is that every $K_5$-minor-free
graph is a minor of a $K_5$-minor-free graph of maximum degree 22.
