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We describe the projective superspace approach to supersymmetric models withoff-shell $(0,4)$ supersymmetry in two dimensions. In addition to the usualsuperspace 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 ourdescription. For instance, it leads to stringent constraints on theinteractions one can write down for a given superfield content of the model. Asan example, we describe in projective superspace Witten's ADHM sigma model -- alinear sigma model with non-derivative interactions whose target is$\mathbf{R}^4$ with a Yang-Mills instanton solution. The hyperk\"ahler natureof target space and the twistor description of instantons by Ward, and Atiyah,Hitchin, Drinfeld and Manin are natural outputs of our construction.

$(0,4)$ Projective Superspaces I: Interacting Linear Sigma Models