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We describe the analysis of the Bergman kernel and projections on worm domain, starting from the breakthrough result of Barrett, following Kiselman, some result by the speaker with $S$. Krantz, up to recent results on the unbounded worm domain $W$. In this case, we show that the Bergman space of $W$ is not trivial. We obtain an asymptotic expansion for $K$ that allows us to describe its singularities at the boundary and to prove the following:
(1) For all $s > 0$, the Bergman projection $P$ does not map the Sobolev space $W^s(W)$ into itself.
(2) For $p$ different from 2, $P$ does not map $L^p(W)$ into itself.
(This is joint work with S. Krantz and C. Stoppato.)