We consider protection problems in multilayer networks. In single-layer networks, a pair of disjoint paths can be used to provide protection for a source-destination pair. However, this approach cannot be directly applied to layered networks where disjoint paths may not always exist. In this paper, we take a new approach which is based on finding a set of paths that may not be disjoint but together will survive any single physical link failure. First, we consider the problem of finding the minimum number of survivable paths. In particular, we focus on two versions of this problem: one where the length of a path is restricted, and the other where the number of paths sharing a fiber is restricted. We prove that in general, finding the minimum survivable path set is NP-hard, whereas both of the restricted versions of the problem can be solved in polynomial time. We formulate the problem as Integer Linear Programs (ILPs), and use these formulations to develop heuristics and approximation algorithms. Next, we consider the problem of finding a set of survivable paths that uses the minimum number of fibers. We show that this problem is NP-hard in general, and develop heuristics and approximation algorithms with provable approximation bounds. Finally, we present simulation results comparing the different algorithms.