In this dissertation we study properties of superconformal field theories (SCFTs) that arise from a variety of constructions. We begin with an extended review of various techniques in supersymmetry that are relevant throughout the work. In Chapter 3, we discuss aspects of theories with superpotentials given by Arnold's $A,D,E$ singularities, particularly the novelties that arise when the fields are matrices. We focus on four-dimensional $N=1$ variants of supersymmetric QCD, with $U(N_c)$ or $SU(N_c)$ gauge group, $N_f$ fundamental flavors, and adjoint matter fields $X$ and $Y$ appearing in $W_{A,D,E}(X,Y)$ superpotentials. We explore these issues by considering various deformations of the $W_{A,D,E}$ superpotentials, and the resulting RG flows and IR theories. In Chapter 4, we examine the infrared fixed points of four-dimensional $N=1$ supersymmetric $SU(2)$ gauge theory coupled to an adjoint and two fundamental chiral multiplets. We focus on a particular RG flow that leads to the $N=2$ Argyres-Douglas theory $H_0$, and a further deformation to an $N=1$ SCFT with low $a$ central charge. Then for the latter half of the dissertation we turn our attention to 4d SCFTs that arise from compactifications of M5-branes. In Chapter 6, we field-theoretically construct 4d $N=1$ quantum field theories by compactifying the 6d (2,0) theories on a Riemann surface with genus $g$ and $n$ punctures, where the normal bundle decomposes into a sum of two line bundles with possibly negative degrees $p$ and $q$. In Chapter 7, we study the 't Hooft anomalies of the SCFTs that arise from these compactifications. In general there are two independent contributions to the anomalies: there is a bulk term obtained by integrating the anomaly polynomial of the world-volume theory on the M5-branes over the Riemann surface, and there is a set of contributions due to local data at the punctures. Using anomaly inflow in M-theory, we describe how this general structure arises for cases when the four-dimensional theories preserve $N=2$ supersymmetry, and derive terms that account for the local data at the punctures.