0 Introduction
1 Electromagnetic duality and monopoles
 1.1 Electric and magnetic charges
 1.2 The S and the T transformations
 1.3 ’t Hooft-Polyakov monopoles
  1.3.1 Classical features
  1.3.2 Semiclassical features
2 $\mathcal{𝒩}=2$ multiplets and Lagrangians
 2.1 Microscopic Lagrangian
  2.1.1 $\mathcal{𝒩}=1$ superfields
  2.1.2 Vector multiplets and hypermultiplets
 2.2 Vacua
 2.3 BPS bound
 2.4 Low energy Lagrangian
3 Renormalization and anomaly
 3.1 Renormalization
 3.2 Anomalies
  3.2.1 Anomalies of global symmetry
  3.2.2 Anomalies of gauge symmetry
 3.3 $\mathcal{𝒩}=1$ pure Yang-Mills
  3.3.1 Confinement and gaugino condensate
  3.3.2 The theory in a box
4 Seiberg-Witten solution to pure $SU\left(2\right)$ theory
 4.1 One-loop running and the monodromy at infinity
 4.2 Behavior in the strongly-coupled region
 4.3 The Seiberg-Witten solution
  4.3.1 The curve
  4.3.2 The monodromy $M_{\infty }$
  4.3.3 The monodromies $M_{\pm }$
 4.4 Less supersymmetric cases
  4.4.1 $\mathcal{𝒩}=1$ system
  4.4.2 Pure bosonic system
 4.5 $SU\left(2\right)$ vs $SO\left(3\right)$
5 $SU\left(2\right)$ theory with one flavor
 5.1 Structure of the $u$-plane
  5.1.1 Schematic running of the coupling
  5.1.2 Monodromies
 5.2 The curve
 5.3 Some notable features
6 Curves and 6d $\mathcal{𝒩}=\left(2,0\right)$ theory
 6.1 Strings with variable tension
 6.2 Strings with variable tension from membranes
  6.2.1 General idea
  6.2.2 Example: pure $SU\left(2\right)$ theory
 6.3 Self-duality of the 6d theory
 6.4 Intermediate 5d Yang-Mills theory and its boundary conditions
  6.4.1 Five-dimensional maximally-supersymmetric Yang-Mills
  6.4.2 $\mathcal{𝒩}=4$ super Yang-Mills
  6.4.3 $\mathcal{𝒩}=2$ pure $SU\left(2\right)$ theory and the $N_f=1$ theory
  6.4.4 The $SU\left(2\right)$ theories with $N_f=2,3,4$
7 Higgs branches and hyperkähler manifolds
 7.1 General structures of the Higgs branch Lagrangian
 7.2 Hypermultiplets revisited
 7.3 The hyperkähler quotient
  7.3.1 $U\left(1\right)$ gauge theory with one charged hypermultiplet
  7.3.2 $SU\left(2\right)$ gauge theory with two hypermultiplets in the doublet
8 $SU\left(2\right)$ theory with 2 and 3 flavors
 8.1 Generalities
 8.2 $N_f=2$: the curve and the monodromies
 8.3 $N_f=2$: the discrete R-symmetry
 8.4 $N_f=2$: the moduli space
 8.5 $N_f=3$
9 $SU\left(2\right)$ theory with 4 flavors and Gaiotto’s duality
 9.1 The curve as ${\lambda }^2={\varphi }_2\left(z\right)$
 9.2 Identification of parameters
  9.2.1 Coupling constant
  9.2.2 Mass parameters
 9.3 Weak-coupling limit and trifundamentals
 9.4 Strong-coupling limit
 9.5 Generalization
  9.5.1 Trivalent diagrams
  9.5.2 Example: torus with one puncture
  9.5.3 Example: sphere with five punctures
  9.5.4 Example: a genus-two surface
  9.5.5 The curve and the Hitchin field
 9.6 Theories with less flavors revisited
  9.6.1 Rewriting of the curves
  9.6.2 Generalization
10 Argyres-Douglas CFTs
 10.1 $N_f=1$ theory and the simplest Argyres-Douglas CFT
 10.2 Argyres-Douglas CFT from the $N_f=2$ theory
 10.3 Argyres-Douglas CFT from the $N_f=3$ theory
 10.4 Summary of rank-1 theories
  10.4.1 Argyres-Douglas CFTs from $SU\left(2\right)$ with flavors
  10.4.2 Exceptional theories of Minahan-Nemeschansky
  10.4.3 Newer rank-1 theories
 10.5 More general Argyres-Douglas CFTs: $X_N$ and $Y_N$
11 Theories with other simple gauge groups
 11.1 Semiclassical analysis
 11.2 Pure $SU\left(N\right)$ theory
  11.2.1 The curve
  11.2.2 Infrared gauge coupling matrix
 11.3 $SU\left(N\right)$ theory with fundamental flavors
  11.3.1 $N_f=1$
  11.3.2 General number of flavors
 11.4 $SO\left(2N\right)$ theories
  11.4.1 Semi-classical analysis
  11.4.2 Pure $SO\left(2N\right)$ theory
  11.4.3 $SO\left(2N\right)$ theory with flavors in the vector representation
 11.5 Argyres-Douglas CFTs
  11.5.1 Pure $SU\left(N\right)$ theory
  11.5.2 $SU\left(N\right)$ theory with two flavors
  11.5.3 Pure $SO\left(2N\right)$ theory
  11.5.4 Argyres-Douglas CFTs and the Higgs branch
 11.6 Seiberg-Witten solutions for various other simple gauge groups
12 Argyres-Seiberg-Gaiotto duality for $SU\left(N\right)$ theory
 12.1 S-dual of $SU\left(N\right)$ with $N_f=2N$ flavors, part I
  12.1.1 Rewriting of the curve
  12.1.2 Weak-coupling limit
  12.1.3 A strong-coupling limit
 12.2 $SU\left(N\right)$ quiver theories and tame punctures
  12.2.1 Quiver gauge theories
  12.2.2 $\mathcal{𝒩}=2^{\ast }$ theory
  12.2.3 Linear quiver theories
  12.2.4 Tame punctures
  12.2.5 Tame punctures and the number of Coulomb branch operators
  12.2.6 Tame punctures and the decoupling
 12.3 S-dual of $SU\left(N\right)$ with $N_f=2N$ flavors, part II
  12.3.1 For general $N$
  12.3.2 $N=3$: Argyres-Seiberg duality
 12.4 Applications
  12.4.1 $T_N$
  12.4.2 $MN\left(E_7\right)$
  12.4.3 $MN\left(E_8\right)$
  12.4.4 The singular limit of $SU\left(N\right)$ with even number of flavors
 12.5 Tame punctures and Higgsing
13 Conclusions and further directions