In this dissertation we discuss some novel applications of the methods of
scattering amplitudes and topological phases. First, we describe on-shell tools
to calculate anomalous dimensions in effective field theories with
higer-dimension operators. Using such tools we prove and apply a new
perturbative non-renormalization theorem, and we explore the structure of the
two-loop anomalous dimension matrix of dimension-six operators in the Standard
Model Effective Theory (SMEFT). Second, we introduce new methods to calculate
the classical limit of gravitational scattering amplitudes. Using these
methods, in conjunction with eikonal techniques, we calculate the classical
gravitational deflection angle of massive and massles particles in a variety of
theories, which reveal graviton dominance beyond 't Hooft's. Finally, we point
out that different choices of Gliozzi-Scherk-Olive (GSO) projections in
superstring theory can be conveniently understood by the inclusion of fermionic
invertible topological phases, or equivalently topological superconductors, on
the worldsheet. We explain how the classification of fermionic
topological phases, recently achieved by the condensed matter community,
provides a complete and systematic classification of ten-dimensional
superstrings and gives a new perspective on the K-theoretic classification of
D-branes.