Preface -- 1 Introduction -- 1.1 Vorticity Dynamics -- 1.2 Hamiltonian Dynamics -- 1.3 Summary of Basic Questions -- 1.4 Exercises -- 2 N Vortices in the Plane -- 2.1 General Formulation -- 2.2 N = 3 -- 2.3 N = 4 -- 2.4 Bibliographic Notes -- 2.5 Exercises -- 3 Domains with Boundaries -- 3.1 Greenโ{128}{153}s Function of the First Kind -- 3.2 Method of Images -- 3.3 Conformai Mapping Techniques -- 3.4 Breaking Integrability -- 3.5 Bibliographic Notes -- 3.6 Exercises -- 4 Vortex Motion on a Sphere -- 4.1 General Formulation -- 4.2 Dynamics of Three Vortices -- 4.3 Phase Plane Dynamics -- 4.4 3-Vortex Collapse -- 4.5 Stereographic Projection -- 4.6 Integrable Streamline Topologies -- 4.7 Boundaries -- 4.8 Bibliographic Notes -- 4.9 Exercises -- 5 Geometric Phases -- 5.1 Geometric Phases in Various Contexts -- 5.2 Phase Calculations For Slowly Varying Systems -- 5.3 Definition of the Adiabatic Hannay Angle -- 5.4 3-Vortex Problem -- 5.5 Applications -- 5.6 Exercises -- 6 Statistical Point Vortex Theories -- 6.1 Basics of Statistical Physics -- 6.2 Statistical Equilibrium Theories -- 6.3 Maximum Entropy Theories -- 6.4 Nonequilibrium Theories -- 6.5 Exercises -- 7 Vortex Patch Models -- 7.1 Introduction to Vortex Patches -- 7.2 The Kida-Neu Vortex -- 7.3 Time-Dependent Strain -- 7.4 Melander-Zabusky-Styczek Model -- 7.5 Geometric Phase for Corotating Patches -- 7.6 Viscous Shear Layer Model -- 7.7 Bibliographic Notes -- 7.8 Exercises -- 8 Vortex Filament Models -- 8.1 Introduction to Vortex Filaments and the LIE -- 8.2 DaRios-Betchov Intrinsic Equations -- 8.3 Hasimotoโ{128}{153}s Transformation -- 8.4 LIA Invariants -- 8.5 Vortex-Stretching Models -- 8.6 Nearly Parallel Filaments -- 8.7 The Vorton Model -- 8.8 Exercises -- References