The general aim of the present monograph is to study boundary-value problems for second-order elliptic operators in Lipschitz subdomains of Riemannian manifolds. In the first part it develops a theory for Cauchy type operators on Lipschitz submanifolds of codimension one (focused on boundedness properties and jump relations). The solution is represented in the form of layer potentials and optimal nontangential maximal function estimates are established. This analysis is carried out under smoothness assumptions (for the coefficients of the operator, metric tensor and the underlying domain) which are in the nature of best possible. In the second part of the monograph, the authors further specialize this discussion to the case of Hodge Laplacian. This time, the goal is to identify all (pairs of) natural boundary conditions of Neumann type. Owing to the structural richness of the higher degree case we are considering, the theory developed here encompasses in a unitary fashion many basic PDEs of mathematical physics. Its scope extends to also cover Maxwell's equations, dealt with separately. The main tools are those of PDEs and harmonic analysis, occasionally supplemented with some basic facts from algebraic topology and differential geometry.