Abstract. We study the irreducible components of the moduli space of instanton sheaves on P 3 , that is rank 2 torsion free sheaves E with c 1 (E) = c 3 (E) = 0 satisfying h 1 (E(−2)) = h 2 (E(−2)) = 0. In particular, we classify all instanton sheaves with c 2 (E) ≤ 4, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space T (d) of stable sheaves on P 3 with Hilbert polynomial P (t) = dt, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity d; we describe all the irreducible components of T (d) for d ≤ 4.