We use a new panoramic imaging survey conducted with the Dark Energy Camera to map the stellar fringes of the Large and Small Magellanic Clouds (LMC/SMC) to extremely low surface brightness V ≈ 32 mag arcsec−2. Our results starkly illustrate the closely interacting nature of the LMC–SMC pair. We show that the outer LMC disk is strongly distorted, exhibiting an irregular shape, evidence for warping, and significant truncation on the side facing the SMC. Large diffuse stellar substructures are present both to the north and south of the LMC, and in the inter-Cloud region. At least one of these features appears as co-spatial with the bridge of RR Lyrae stars that connects the Clouds. The SMC is highly disturbed; we confirm the presence of tidal tails, as well as a large line-of-sight depth on the side closest to the LMC. Young, intermediate-age, and ancient stellar populations in the SMC exhibit strikingly different spatial distributions. In particular, those with ages ~1.5–4 Gyr exhibit a spheroidal distribution with a centroid offset from that of the oldest stars by several degrees toward the LMC. We speculate that the gravitational influence of the LMC may already have been perturbing the gaseous component of the SMC several Gyr ago. With careful modeling, the variety of substructures and tidal distortions evident in the Magellanic periphery should tightly constrain the interaction history of the Clouds.