We consider the problem of learning the weighted edges of a graph by observing the noisy times of infection for multiple epidemic cascades on this graph. Past work has considered this problem when the cascade information, i.e., infection times, are known exactly. Though the noisy setting is well motivated by many epidemic processes (e.g., most human epidemics), to the best of our knowledge, very little is known about when it is solvable. Previous work on the no-noise setting critically uses the ordering information. If noise can reverse this -- a node's reported (noisy) infection time comes after the reported infection time of some node it infected -- then we are unable to see how previous results can be extended. We therefore tackle two versions of the noisy setting: the limited-noise setting, where we know noisy times of infections, and the extreme-noise setting, in which we only know whether or not a node was infected. We provide a polynomial time algorithm for recovering the structure of bidirectional trees in the extreme-noise setting, and show our algorithm almost matches lower bounds established in the no-noise setting, and hence is optimal up to log-factors. We extend our results for general degree-bounded graphs, where again we show that our (poly-time) algorithm can recover the structure of the graph with optimal sample complexity. We also provide the first efficient algorithm to learn the weights of the bidirectional tree in the limited-noise setting. Finally, we give a polynomial time algorithm for learning the weights of general bounded-degree graphs in the limited-noise setting. This algorithm extends to general graphs (at the price of exponential running time), proving the problem is solvable in the general case. All our algorithms work for any noise distribution, without any restriction on the variance.