For $G$ a compactLie group there is a way to equip the topological classifying space$B G$ with smooth structure such that the corresponding smooth universal principal bundle$E G \to B G$ carries a smooth connection$\nabla_{univ}$ with the property that for every $G$-principal bundle$P \to X$ with connection$\nabla$ there is a smooth representative $f : X \to B G$ of the classifying map, such that $(P, \nabla) \simeq (P, f^* \nabla_{univ})$. This $\nabla_{univ}$ is called the universal $G$-connection.