1. An Introduction to Vortex Dynamics for Incompressible Fluid Flows. 1.1. The Euler and the Navier-Stokes Equations. 1.2. Symmetry Groups for the Euler and the Navier-Stokes Equations. 1.3. Particle Trajectories. 1.4. The Vorticity, a Deformation Matrix, and Some Elementary Exact Solutions. 1.5. Simple Exact Solutions with Convection, Vortex Stretching, and Diffusion. 1.6. Some Remarkable Properties of the Vorticity in Ideal Fluid Flows. 1.7. Conserved Quantities in Ideal and Viscous Fluid Flows. 1.8. Leray's Formulation of Incompressible Flows and Hodge's Decomposition of Vector Fields -- 2. The Vorticity-Stream Formulation of the Euler and the Navier-Stokes Equations. 2.1. The Vorticity-Stream Formulation for 2D Flows. 2.2. A General Method for Constructing Exact Steady Solutions to the 2D Euler Equations. 2.3. Some Special 3D Flows with Nontrivial Vortex Dynamics. 2.4. The Vorticity-Stream Formulation for 3D Flows. 2.5. Formulation of the Euler Equation as an Integrodifferential Equation for the Particle Trajectories -- 3. Energy Methods for the Euler and the Navier-Stokes Equations. 3.1. Energy Methods: Elementary Concepts. 3.2. Local-in-Time Existence of Solutions by Means of Energy Methods. 3.3. Accumulation of Vorticity and the Existence of Smooth Solutions Globally in Time. 3.4. Viscous-Splitting Algorithms for the Navier-Stokes Equation -- 4. The Particle-Trajectory Method for Existence and Uniqueness of Solutions to the Euler Equation. 4.1. The Local-in-Time Existence of Inviscid Solutions. 4.2. Link between Global-in-Time Existence of Smooth Solutions and the Accumulation of Vorticity through Stretching. 4.3. Global Existence of 3D Axisymmetric Flows without Swirl. 4.4. Higher Regularity -- 5. The Search for Singular Solutions to the 3D Euler Equations. 5.1. The Interplay between Mathematical Theory and Numerical Computations in the Search for Singular Solutions. 5.2. A Simple 1D Model for the 3D Vorticity Equation. 5.3. A 2D Model for Potential Singularity Formation in 3D Euler Equations. 5.4. Potential Singularities in 3D Axisymmetric Flows with Swirl. 5.5. Do the 2D Euler Solutions Become Singular in Finite Times? -- 6. Computational Vortex Methods. 6.1. The Random-Vortex Method for Viscous Strained Shear Layers. 6.2. 2D Inviscid Vortex Methods. 6.3. 3D Inviscid-Vortex Methods. 6.4. Convergence of Inviscid-Vortex Methods. 6.5. Computational Performance of the 2D Inviscid-Vortex Method on a Simple Model Problem. 6.6. The Random-Vortex Method in Two Dimensions -- 7. Simplified Asymptotic Equations for Slender Vortex Filaments. 7.1. The Self-Induction Approximation, Hasimoto's Transform, and the Nonlinear Schrodinger Equation. 7.2. Simplified Asymptotic Equations with Self-Stretch for a Single Vortex Filament. 7.3. Interacting Parallel Vortex Filaments -- Point Vortices in the Plane. 7.4. Asymptotic Equations for the Interaction of Nearly Parallel Vortex Filaments. 7.5. Mathematical and Applied Mathematical Problems Regarding Asymptotic Vortex Filaments -- 8. Weak Solutions to the 2D Euler Equations with Initial Vorticity in L[superscript [infinity]]. 8.1. Elliptical Vorticies. 8.2. Weak L[superscript [infinity]] Solutions to the Vorticity Equation. 8.3. Vortex Patches -- 9. Introduction to Vortex Sheets, Weak Solutions, and Approximate-Solution Sequences for the Euler Equation. 9.1. Weak Formulation of the Euler Equation in Primitive-Variable Form. 9.2. Classical Vortex Sheets and the Birkhoff-Rott Equation. 9.3. The Kelvin-Helmholtz Instability. 9.4. Computing Vortex Sheets. 9.5. The Development of Oscillations and Concentrations -- 10. Weak Solutions and Solution Sequences in Two Dimensions. 10.1. Approximate-Solution Sequences for the Euler and the Navier-Stokes Equations. 10.2. Convergence Results for 2D Sequences with L[superscript l] and L[superscript p] Vorticity Control -- 11. The 2D Euler Equation: Concentrations and Weak Solutions with Vortex-Sheet Initial Data. 11.1. Weak- and Reduced Defect Measures. 11.2. Examples with Concentration. 11.3. The Vorticity Maximal Function: Decay Rates and Strong Convergence. 11.4. Existence of Weak Solutions with Vortex-Sheet Initial Data of Distinguished Sign -- 12. Reduced Hausdorff Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions. 12.1. The Reduced Hausdorff Dimension. 12.2. Oscillations for Approximate-Solution Sequences without L[superscript l] Vorticity Control. 12.3. Young Measures and Measure-Valued Solutions of the Euler Equations. 12.4. Measure-Valued Solutions with Oscillations and Concentrations -- 13. The Vlasov-Poisson Equations as an Analogy to the Euler Equations for the Study of Weak Solutions. 13.1. The Analogy between the 2D Euler Equations and the 1D Vlasov-Poisson Equations. 13.2. The Single-Component 1D Vlasov-Poisson Equation. 13.3. The Two-Component 1D Vlasov-Poisson System.