A manifold is a mathematical space that locally resembles standard Euclidean space. In order to study the geometry of such a space, it is necessary to prescribe a connection on the manifold. A connection describes how tangent spaces to the manifold change with respect to infinitesimal changes in the manifold. In 1950, Charles Ehresmann defined a connection to be an abstract object called a horizontal bundle, with a special property called horizontal path lifting. By including this additional property in his definition, Ehresmann implicitly acknowledged that it was nontrivial: not all horizontal bundles satisfy it. We give the first complete characterization of the horizontal bundles which have this property, and hence are connections.