  The real and complex zeros of the parabolic cylinder function $U(a,z)$ are
studied. Asymptotic expansions for the zeros are derived, involving the zeros
of Airy functions, and these are valid for $a$ positive or negative and large
in absolute value, uniformly for unbounded $z$ (real or complex). The accuracy
of the approximations of the complex zeros is then demonstrated with some
comparative tests using a highly precise numerical algorithm for finding the
complex zeros of the function.
