  We describe the projective superspace approach to supersymmetric models with
off-shell $(0,4)$ supersymmetry in two dimensions. In addition to the usual
superspace coordinates, projective superspace has extra bosonic variables --
one doublet for each $\text{SU}(2)$ in the R-symmetry $\text{SU}(2) \times
\text{SU}(2)$ which are interpreted as homogeneous coordinates on
$\mathbf{CP}^1 \times \mathbf{CP}^1$. The superfields are analytic in the
$\mathbf{CP}^1$ coordinates and this analyticity plays an important role in our
description. For instance, it leads to stringent constraints on the
interactions one can write down for a given superfield content of the model. As
an example, we describe in projective superspace Witten's ADHM sigma model -- a
linear sigma model with non-derivative interactions whose target is
$\mathbf{R}^4$ with a Yang-Mills instanton solution. The hyperk\"ahler nature
of target space and the twistor description of instantons by Ward, and Atiyah,
Hitchin, Drinfeld and Manin are natural outputs of our construction.
