  We consider three problems connected with coinvariant subspaces of the
backward shift operator in Hardy spaces $H^p$: 1) properties of truncated
Toeplitz operators; 2) Carleson-type embedding theorems for the coinvariant
subspaces; 3) factorizations of pseudocontinuable functions from $H^1$. These
problems turn out to be closely connected and even, in a sense, equivalent. The
new approach based on the factorizations allows us to answer a number of
challenging questions about truncated Toeplitz operators posed by Donald
Sarason.
