  Generalizing the definitions originally presented by Kuznetsov and Faenzi, we
study (possibly non locally free) instanton sheaves of arbitrary rank on Fano
threefolds. We classify rank 1 instanton sheaves and describe all curves whose
structure sheaves are rank 0 instanton sheaves. In addition, we show that every
rank 2 instanton sheaf is an elementary transformation of a locally free
instanton sheaf along a rank 0 instanton sheaf. To complete the paper, we
describe the moduli space of rank 2 instanton sheaves of charge 2 on a quadric
threefold $X$, and show that the full moduli space of rank 2 semistable sheaves
on $X$ with Chern classes $(c_1,c_2,c_3)=(-1,2,0)$ is connected and contains,
besides the instanton component, just one other irreducible component which is
also fully described.
