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)...