Compressions of Toeplitz operators to coinvariant subspaces of $H^2$ are
called \emph{truncated Toeplitz operators}. We study two questions related to
these operators. The first, raised by Sarason, is whether boundedness of the
operator implies the existence of a bounded symbol; the second is the
reproducing kernel thesis. We show that in general the answer to the first
question is negative, and we exhibit some classes of spaces for which the
answers to both questions are positive.