  We study moduli spaces of stable objects in the Kuznetsov components of Fano
threefolds. We prove a general non-emptiness criterion for moduli spaces, which
applies to the cases of prime Fano threefolds of index $1$, degree $10 \leq d
\leq 18$, and index $2$, degree $d \leq 4$. In the second part, we focus on
cubic threefolds. We show the irreducibility of the moduli spaces, and that the
general fibers of the Abel--Jacobi maps from the moduli spaces to the
intermediate Jacobian are Fano varieties. When the dimension is sufficiently
large, we further show that the general fibers of the Abel--Jacobi maps are
stably birational equivalent to each other. As an application of our methods,
we prove Conjecture A.1 in [FGLZ24] concerning the existence of Lagrangian
subvarieties in moduli spaces of stable objects in the Kuznetsov components of
very general cubic fourfolds.
