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We study the irreducible components of the moduli space of instanton sheaveson $\mathbb{P}^3$, 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, weclassify all instanton sheaves with $c_2(E)\le4$, describing all theirreducible components of their moduli space. A key ingredient for our argumentis the study of the moduli space ${\mathcal T}(d)$ of stable sheaves on$\mathbb{P}^3$ with Hilbert polynomial $P(t)=d\cdot t$, which contains, as anopen subset, the moduli space of rank 0 instanton sheaves of multiplicity $d$;we describe all the irreducible components of ${\mathcal T}(d)$ for $d\le4$.

Moduli spaces of rank 2 instanton sheaves on the projective space