In this series of talks, I will first present some recently introduced geometric notions called fiberwise amenability and almost elementariness for locally compact Hausdorff \'{e}tale groupoids and show how they can be used to establish (tracial) $\mathcal{Z}$-stability of groupoid $C^*$-algebras and thus the classifiablity of this kind of $C^*$-algebras. Concrete examples from the literature will be provided. In addition, I will discuss how the fiberwise amenability reflects the soficity of the topological full groups of the groupoid. This then leads to a new framework for studying the amenability of the topological full groups of Cantor dynamical systems. New results on LEF topological full groups will also be addressed. Parts of the research reported here are based on collaborated work with Jianchao Wu.