1 Introduction

2 Preliminaries

3 Classification of Simple and Nonsimple Graph $C^{*}$-algebras

4 In Search of Techniques to Classify Algebras: Shift Spaces, Flow Equivalence, and Moves on Graphs

5 Classification of Leavitt Path Algebras of Finite Graphs

6 Classification of Leavitt Path Algebras of Infinite Graphs