A working knowledge of $\mathcal{\mathcal{N}}=1$ superﬁelds is required; we set up our notation in Sec. 2. Similarly, a reader should know one-loop renormalization and perturbative anomalies, and should have at least heard about instantons and monopoles, although we give a quick summary and references. No prior knowledge of string theory or M-theory is assumed, but a reader should be open to the concept of theories deﬁned in spacetime whose dimension is larger than four. Unless otherwise stated, we use the common physics convention of calling $SU\left(N\right)$ whatever gauge group whose gauge algebra is $\U0001d530\U0001d532\left(N\right)$, etc.

Signs and powers of $i$ in the terms in the Lagrangian are not completely consistent or correct, but the overall ideas presented in the lecture note should be alright. The author is sorry that he used the same letter $i$ for the imaginary unit and for the indices, and the same letter $\mathit{\theta}$ for the theta angle and for the supercoordinates. In general, readers are encouraged to read not just what is written, but what should be written instead. Presumably there are many other typos, errors, and points to be improved. The author would welcome whatever comments from you, so please do not hesitate to write an email to the author at yuji.tachikawa@ipmu.jp.

The deﬁciencies concerning citations are most obvious, as the number of relevant papers is immense. The author is quite sure that he cited too much of his own papers. Other than that, the author at least tried to give a few pointers to recent papers from whose references the interested readers should be able to start exploring the literature. The author is open to add more references in this lecture note itself, and any reader is again encouraged to send emails.

This lecture note is based on the author’s lectures at Nagoya University and Tohoku University on 2011 and at Rikkyo University on 2013. The author thanks the hosts in these three universities for giving him the opportunity to present a review of the $\mathcal{\mathcal{N}}=2$ supersymmetric dynamics using new techniques. He also thanks the participants of these lectures for giving him many useful comments along the course of the lectures. The author’s approach to this topic has been formed and heavily inﬂuenced by the discussions with various colleagues, and most of the new arguments in this note, except for those which are wrong, should not be credited to the author. Ofer Aharony, Lakshya Bhardwaj, Jacques Distler, Satoshi Nawata, Vasily Pestun and Futoshi Yagi gave helpful comments on the draft version of this lecture note. Simone Giacomelli, Brian Henning, Greg Moore, Tatsuma Nishioka, Jun’ya Yagi and Kazuya Yonekura in particular read the draft in detail and suggested many points to be improved to the author. It is a pleasure and indeed a privilege that the author can thank them.

The author also thanks the right amount of duties associated to his position, with which he cannot concentrate any longer on cutting-edge researches but still has some time to summarize what he already knows. In particular, he thanks various stupid faculty meetings he needs to participate, during which time he drew most of the ﬁgures on his laptop. The readers should therefore thank the overly bureaucratic system prevalent in University of Tokyo, which made this lecture note materialize. This work is supported in part by JSPS Grant-in-Aid for Scientiﬁc Research No. 25870159 and in part by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo.