When do the spectra of operators vary continuously? How to show such a continuity if the operators depend on a underlying geometry and dynamics that changes? The theory of continuous fields of C*-algebras has a long history and goes back to the works of Kaplansky and Fell [5, 6, 3, 4]. Already Kaplansky [6] realized that this notion implies continuity of the spectra of self-adjoint elements with respect to the Hausdorff metric. During the talk, we review the connections to the spectra and discuss its applications. In the second part of the talk, we focus on more recent developments [2] controlling the rate of convergence where we will focus on dynamical defined operators. Tools to construct continuous fields of C*-algebras often involve some amenability condition. Also in this recent work the interplay with amenability leads to interesting observations.