  In four-dimensional N=1 Minkowski superspace, general nonlinear sigma models
with four-dimensional target spaces may be realised in term of CCL (chiral and
complex linear) dynamical variables which consist of a chiral scalar, a complex
linear scalar and their conjugate superfields. Here we introduce CCL sigma
models that are invariant under U(1) "duality rotations" exchanging the
dynamical variables and their equations of motion. The Lagrangians of such
sigma models prove to obey a partial differential equation that is analogous to
the self-duality equation obeyed by U(1) duality invariant models for nonlinear
electrodynamics. These sigma models are self-dual under a Legendre
transformation that simultaneously dualises (i) the chiral multiplet into a
complex linear one; and (ii) the complex linear multiplet into a chiral one.
Any CCL sigma model possesses a dual formulation given in terms of two chiral
multiplets. The U(1) duality invariance of the CCL sigma model proves to be
equivalent, in the dual chiral formulation, to a manifest U(1) invariance
rotating the two chiral scalars. Since the target space has a holomorphic
Killing vector, the sigma model possesses a third formulation realised in terms
of a chiral multiplet and a tensor multiplet.
  The family of U(1) duality invariant CCL sigma models includes a subset of
N=2 supersymmetric theories. Their target spaces are hyper Kahler manifolds
with a non-zero Killing vector field. In the case that the Killing vector field
is triholomorphic, the sigma model admits a dual formulation in terms of a
self-interacting off-shell N=2 tensor multiplet.
  We also identify a subset of CCL sigma models which are in a one-to-one
correspondence with the U(1) duality invariant models for nonlinear
electrodynamics. The target space isometry group for these sigma models
contains a subgroup U(1) x U(1).
