In this paper we consider scalar Riemann solvers on networks, associated to scalar conservation laws. A junction is a particular network which is a finite set of half lines glued together at the origin. Riemann solvers solve uniquely the Riemann problem on the junction. We also assume that Riemann solutions are stable by passage to the limit. In part I of the paper, we only address fundamental questions concerning Riemann problems on junctions. We show a characterization of Riemann solvers either by their set of stationary solutions (the germ), or equivalently by their Godunov flux at the junction. Moreover, we show that the gluing of two junctions with Riemann solvers is well defined and leads to a new junction with a new Riemann solver. Because our theory is quite general, it encompasses in particular Kruzkov germs, Hamilton-Jacobi germs, monotone germs, conservative and non-conservative germs. In part II of the paper, we give an existence and uniqueness theory for conservation laws on networks in the special case where Riemann solvers are associated to Kruzkov germs.