  We define a regularized Shintani theta lift which maps weight $2k+2$ ($k \in
\Z, k \geq 0$) harmonic Maass forms for congruence subgroups to
(sesqui-)harmonic Maass forms of weight $3/2+k$ for the Weil representation of
an even lattice of signature $(1,2)$. We show that its Fourier coefficients are
given by traces of CM values and regularized cycle integrals of the input
harmonic Maass form. Further, the Shintani theta lift is related via the
$\xi$-operator to the Millson theta lift studied in our earlier work. We use
this connection to construct $\xi$-preimages of Zagier's weight $1/2$
generating series of singular moduli and of some of Ramanujan's mock theta
functions.
