4D $N=$ 2 SCFTs and spin chains

This is the writeup of the lectures given at the Winter School 'YRISW 2018' to appear in a special issue of J. Phys. A: Math. Theor. In the first part of these lecture notes we review some important facts about 4D $N=2$ SCFTs. We begin with basic textbook material, the supersymmetry algebra and its massless representations and the construction of Lagrangians using superspace. Then we turn to more modern topics, the study of the $N=2$ SCA and its representation theory. Our intention is to understand how much we can learn from representation theory alone, even about the dynamics of $N=2$ SCFTs. In the second part of the notes we use these tools to construct spin chains for $N=2$ SCFTs, the spectral problem of which computes anomalous dimensions of local operators. We discuss their novel features comparing them with their counterparts in $N=4$ SYM and search for possible integrability structures that emerge. (topical review)