In this talk, I will give develop the homotopy theory of Batalin-Vilkovisky algebras and give several applications: resolution of the Lian-Zuckerman conjecture (vertex algebras), description of the stabilizers of cohomological field theories under Givental group action (intersection theory of moduli spaces of curves), homotopy trivialization of the action of the circle on the framed little disks operad and homotopy Frobenius manifold structure on the de Rham cohomology of Poisson manifolds (mirror symmetry conjecture).