This paper considers a fully general (Riemann) wave equation on a finite-dimensional Riemannian manifold, with energy level (H-1 x L-2) -terms, under essentially minimal smoothness assumptions on the variable (in time and space) coefficients. The paper provides Carleman-type inequalities: first pointwise, for C-2- solutions, then in integral form for H-1,H-1(Q)-solutions. The aim of the present approach is to provide Carleman inequalities which do not contain lower-order terms, a distinguishing feature over most of the literature. Accordingly, global uniqueness results for overdetermined problems as well as Continuous Observability/Uniform Stabilization inequalities follow in one shot, as a part of the same stream of arguments. Constants in the estimates are, therefore, generally explicit. The paper emphasizes the more challenging pure Neumann B.C. case. The paper is a generalization from the Euclidean to the Riemannian setting of [LTZ] in the more difficult case of purely Neumann B.C., and of [KK1] in the case of Dirichlet B.C. The approach is Riemann geometric, but different from-indeed, more flexible than-the one in [LTY1].