  We study the sensitivity of possible CLIC and FCC-ee measurements of
light-by-light scattering to old and new physics, including the
Heisenberg-Euler Lagrangian in the Standard Model with possible contributions
from loops of additional charged particles or magnetic monopoles, the
Born-Infeld extension of QED, and effective dimension-8 operators involving
four electromagnetic field strengths as could appear in the Standard Model
Effective Field Theory. We find that FCC-ee measurements at 365 GeV and CLIC
measurements at 350 GeV would be sensitive to new physics scales of half a TeV
in the dimension-8 operator coefficients, and that CLIC measurements at 1.4 TeV
or 3 TeV would be sensitive to new physics scales {$\sim 2$ TeV or 5 TeV} at
95\% CL, corresponding to probing loops of new particles with masses up to
$\sim 3.7$ TeV {for large charges and/or multiple species}. Within Born-Infeld
theory, the $95\%$ CL sensitivities would range from $\sim 300$ GeV to 1.3 or
2.8 TeV for the high-energy CLIC options. Measurements of light-by-light
scattering would not exclude monopole production at FCC-hh, except in the
context of Born-Infeld theory.
