We construct a notion of parallel transport along superpaths in a manifold that corresponds to a superconnection (`{a} la Quillen), in an attempt to understand geometrically superconnections, the same way as an appropriate notion of parallel transport along paths translates geometrically the concept of a connection. The parallel transport along superpaths is realized by solving some ``half-order' differential equations, as opposed to solving first-order differential equations for the usual parallel transport. Before doing this, we extend the usual notion of parallel transport along paths associated to a connection to superpaths, and see how the super-parallel transport incorporates the analytical concept of a connection. Such considerations are motivated by trying to understand one dimensional supersymmetric field theories over a manifold, in the hope that they provide geometric cocycles for differential K-theory. The larger context is the Stolz-Teichner program (see cite{ST}) of relating field theories and cohomology theories, and our effort is to complete the understanding of the one-dimensional story.