The celebrated Witten conjecture initiates the study of relation between intersection theories on moduli spaces of curves (axiomatized as cohomological field theories, CohFTs) and integrable hierarchies. In this talk we will briefly review generalizations of the Witten conjecture over decades. Then we introduce our recent result relating CohFTs with the KP integrable hierarchies, via the technique of topological recursion on spectral curves. It is based on the joint work with Shuai Guo and Qingsheng Zhang.