  The Euclidean path integral is well approximated by instantons. If instantons
are dynamical, then instantons are necessarily complexified. These fuzzy
instantons can have various physical applications. In slow-roll inflation
models, fuzzy instantons can explain the probability distribution of the
initial conditions of the universe. Even though the potential shape does not
satisfy the slow-roll conditions following the swampland criteria, still the
fuzzy instantons can explain the origin of the universe. If we extend the
Euclidean path integral beyond the no-boundary proposal, we may study fuzzy
Euclidean wormholes that have various physical applications in cosmology and
black hole physics. We summarize them and discuss possible future research
topics.
