  In this paper, we study Bloch's conjecture for zero cycles on K3 surfaces and
hyper-K\"ahler varieties. We prove Bloch's conjecture for reflexive
autoequivalences on K3 surfaces. This confirms Bloch's conjecture for all
(anti)-symplectic autoequivalences of a K3 surface with Picard number $>2$. As
a consequence, we prove Bloch's conjecture for (anti)-symplectic birational
automorphisms on Bridgeland moduli space of a K3 surface with Picard number $>
2$. Furthermore, we generalize Huybrechts' work in \cite{Huy12} to twisted K3
surfaces which asserts that the action of autoequivalences of a twisted K3
surface on the Chow group is determined by its action on the cohomology group.
This allows us to prove Bloch's conjecture for symplectic birational
automorphisms on arbitrary $K3^{[n]}$-type hyper-K\"ahler varieties preserving
a birational Lagrangian fibration. We also establish an anti-symplectic version
provided Voisin's filtration is the same as the filtration introduced in
\cite{SYZ20}. Finally, we prove the constant cycle property for the fixed loci
of anti-symplectic involutions on hyper-K\"ahler variety of $K3^{[n]}$-type if
$n\leq 2$ or the N\'eron-Severi invariant sublattice has rank $>1$.
