In this PhD dissertation, we study epidemics on networks of contacts through the lens of statistical inference. The current work is an attempt to infer the propagation parameters following the outset of an epidemic spread. My contributions rely on the progress on mathematical modeling of infectious outbreak, information diffusion, and viral habit formation. These achievements paved the path to forecast and contain the spread of infectious diseases and to optimize viral marketing campaigns. What distinguishes this work is the forensics view that aims to infer the network or the propagation parameters from the final stage of an epidemic. We study here multiple problems of this kind including epidemic source identification and epidemic network reconstruction. Such problems are NP-hard by nature and previous contributions are ad-hoc and inconclusive for realistic networks, either in size or structure. This work proposes new methods that estimate the parameters of interest in polynomial time with arbitrary accuracy. We provide theoretical error bound guarantees for some of the solutions. We accompany the results with comparative simulations on popular networks from social media, urban infrastructure, and disease pandemics.