Noncommutative tori are important examples of noncommutative spaces. Following seminal work by Connes-Tretkoff, Connes-Moscovici, Fathizadeh-Khalkhali, and others a differential geometric apparatus on NC tori is currently being built. So far the main focus has been mostly on conformal deformation of the (flat) Euclidean metric or product of such metrics. A new challenge due to the extra noncommutativity is the accounting of the non-triviality of the modular automorphism groups of the weights at stake. This talk will report on ongoing work to deal with general Riemannian metrics on NC tori (in the sense of Jonathan Rosenberg). After recalling the construction of the Laplace-Beltrami operator in this setting, several results will be presented. The first set of results concern versions for NC tori of Connes' trace theorem and Connes' integration formula. This sheds new light on scalar curvature for NC tori. The other set of result concerns versions of Gauss-Bonnet theorems for arbitrary Riemannian metrics. This encapsulates various versions of Gauss-Bonnet theorems that have been established for special classes of metrics.