This paper studies online shortest path routing over dynamic multi-hop networks. Link costs or delays are time-varying and modelled by independent and identically distributed random processes, whose parameters are initially unknown. The parameters, and hence the optimal path, can only be estimated by routing packets through the network and observing the realized delays. Our aim is to find a routing policy that minimizes the regret (the cumulative delay difference) between the path chosen by the policy and the unknown optimal path. We formulate the problem as a combinatorial bandit optimization problem and consider several scenarios that differ in where routing decisions are made and in the information available when making the decision. For each scenario, we derive the tight asymptotic lower bound on the regret that has to be satisfied by any online routing policy. These bounds help us to understand the performance improvements we can expect when (i) taking routing decisions at each hop rather than at the source only, and (ii) observing per-link costs rather than aggregate path costs. In particular, we show that (i) is of no use while (ii) can have a spectacular impact. Efficient algorithms are proposed and evaluated against the state-of-the art.