To avoid the technicalities of unbounded operators and their dense domains, in this talk I will deal only with finite-dimensional C*-algebras. I will introduce what I am calling a Riemannian metric over such an algebra A. When A is commutative I will indicate how we essentially obtain a (finite) resistance network. I will describe interesting non-commutative examples. In particular, in our setting every spectral triple determines a Riemannian metric. I will sketch how from a Riemannian metric we obtain further interesting structures, such as Laplace operators, seminorms equipping A with the structure of a quantum metric space, and corresponding metrics on the state space. These seminorms have surprisingly strong properties. I will also mention how this setup is closely related to Dirichlet forms and quantum semigroups.