I will discuss the large N behavior of partition functions of the ABJM theory on compact Euclidean manifolds. I will pay particular attention to the S^3 free energy and the topologically twisted index for which I will present closed form expressions valid to all orders in the large N expansion. These results have important implications for holography and the microscopic entropy counting of AdS_4 black holes which I will discuss. I will also briefly discuss generalizations to other SCFTs arising from M2-branes.

I will describe a class of supergravity solutions holographically dual to d-dimensional maximally supersymmetric SYM on S^d. Supersymmetric localization can be employed to calculate the partition function and the VEV of a 1/2-BPS Wilson lines in the planar limit of the SYM theory. I will present the results of this calculation and will show how they lead to a non-trivial precision test of holography in the context of non-conformal QFTs and space-times that are non asymptotically locally AdS.

I will describe the one-dimensional complex conformal manifold that controls the infrared dynamics of a three-dimensional N=2 supersymmetric theory of three chiral superfields with a cubic superpotential. Two special points on this conformal manifold are the XYZ model and three decoupled copies of the critical Wess-Zumino model. The conformal manifold enjoys a discrete duality group and can be thought of as an orbifold of CP^1. I will discuss how to compute the spectrum of low lying operators and their OPE coefficients as a function of the coordinate on the conformal manifold using the epsilon-expansion and the numerical conformal bootstrap.

Superconformal field theories placed in nontrivial background fields for the metric and the continuous global symmetries exhibit a rich web of RG flows across dimensions. I will discuss several examples of such flows and emphasize some of their universal features. In addition, I will employ non-perturbative tools such as 't Hooft anomaly matching, a-, F-, and c-extremization, and holography to gain a quantitative understanding of some aspects of these theories. Finally, I will discuss the relevance of these results for a microscopic understanding of the entropy of supersymmetric black holes and strings in AdS.

I will discuss the gravitational dual of a mass deformation of N=4 SYM, called N=2* SYM, on S^4. Using holographic techniques one can calculate the universal contribution to the corresponding free energy in the planar limit and at large 't Hooft coupling. The result matches the expression recently computed using supersymmetric localization in the field theory. This agreement constitutes a non-trivial precision test of holography in a non-conformal setting. I will also briefly discuss the extension of these results to mass deformations of N=4 SYM with N=1 supersymmetry.