—Network Tomography (or network monitoring) uses end-to-end path-level measurements to characterize the network, such as topology estimation and failure detection. This work provides the first comprehensive study of passive network tomography in the presence of network failures under the setting that all nodes perform random linear network coding. In particular, we show that it is both necessary and sufficient for all nodes in the network to share common randomness, i.e., all local coding coefficients are chosen using a commonly shared random code-book. Without such common randomness, we prove that in the presence of adversarial or random failures, it is either theoretically impossible or computationally intractable to accurately estimate the topology of general networks, and then locate the failures. With such common randomness, we present several sets of algorithms for topology estimation and failure detection, under various settings of adversarial and random failures. For some ...