  In these lectures I talk about simplifications and universalities found in
scattering amplitudes for gauge and gravity theories. In contrast to Ward
identities, which are understood to arise from familiar symmetries of the
classical action, these structures are currently only understood in terms of
graphical organizational principles, such as the gauge-theoretic
color-kinematics duality and the gravitational double-copy structure, for local
representations of multi-loop S-matrix elements. These graphical principles
make manifest new relationships in and between gauge and gravity scattering
amplitudes. My lectures will focus on arriving at such graphical organizations
for generic theories with examples presented from maximal supersymmetry, and
their use in unitarity-based multi-loop integrand construction.
