  The DT-operators are introduced, one for every pair (\mu,c) consisting of a
compactly supported Borel probability measure \mu on the complex plane and a
constant c>0. These are operators on Hilbert space that are defined as limits
in *-moments of certain upper triangular random matrices. The DT-operators
include Voiculescu's circular operator and elliptic deformations of it, as well
as the circular free Poisson operators. We show that every DT-operator is
strongly decomposable. We also show that a DT-operator generates a II_1-factor,
whose isomorphism class depends only on the number and sizes of atoms of \mu.
Those DT-operators that are also R-diagonal are identified. For a
quasi-nilpotent DT-operator T, we find the distribution of T^*T and a recursion
formula for general *-moments of T.
