  Recently N.Nekrasov and A.Schwarz proposed a modification of the ADHM
construction of instantons which produces instantons on a noncommutative
deformation of the 4-dimensional real affine space. In this paper we study the
relation between their construction and algebraic bundles on noncommutative
projective spaces. We exhibit one-to-one correspondences between three classes
of objects: framed bundles on a noncommutative projective plane, certain
complexes of sheaves on a noncommutative 3-dimensional projective space, and
the modified ADHM data. The modified ADHM construction itself is interpreted in
terms of a noncommutative version of the twistor transform. We also prove that
the moduli space of framed bundles on the noncommutative projective plane has a
natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the
moduli space of framed torsion free sheaves on the commutative projective
plane. The natural complex structures on the two moduli spaces do not coincide
but are related by an SO(3) rotation. Finally, we propose a construction of
instantons on a more general noncommutative R^4 than the one considered by
Nekrasov and Schwarz (a q-deformed R^4).
