Black hole and cosmological horizons play
a crucial role in physics. They are central to our understanding of the origin
of structure in the universe, while continuing to provide vexing theoretical
puzzles. They have become accessible observationally to a remarkable degree,
albeit indirectly. I will review how horizons appear in general relativity and
quantum field theory. Then I will move to a systematic study of their breakdown
and its relevance -- or more precisely, `dangerous irrelevance' -- to
thought experiments and real observations in specific situations. After
describing the sensitivity of primordial cosmological perturbations to heavy
degrees of freedom and large field values, I will share some new results
exhibiting the breakdown of effective quantum field theory for string-theoretic
probes of black hole horizons.