The present dissertation thesis is concerned with Banach space-valued convergence theorems along Folner type sequences in geometries with amenable structures. The text contains new ergodic theorems for groups, as well as an almost-additive and a subadditive convergence theorem for hyperfinite, weakly convergent graph sequences. All these results - among them abstract mean and pointwise ergodic theorems - significantly extend the literature on ergodic theorems. As an application, one obtains the uniform convergence of the integrated density of states (IDS) for pattern-invariant, finite hopping range operators in a wide range of amenable situations. A different kind of spectral approximation is given by a strong result for Ihara's Zeta function associated with sofic graphings. It is shown that compact convergence can be obtained for the normalized finite versions corresponding to the elements in a weakly convergent graph sequence.