We prove the existence and multiplicity of solutions for a class of Choquard-Kirchhoff type equations with variable exponents and critical reaction. Because the appearance of the critical reaction, we deal with the lack of compactness by using the concentration-compactness principle. In particular, we discuss the main results in non-degenerate and degenerate cases. And we apply combination of Krasnoselskii genus and the Hardy-Littlewood-Sobolev inequality to get the results of existence and multiplicity.