In this article, the cluster-in-molecule (CIM) local correlation approach for periodic systems with periodic boundary condition has been developed, which allows electron-correlation calculations of various crystals computationally tractable. In this approach, the correlation energy per unit cell of a periodic system can be evaluated as the summation of the correlation contributions from electron-correlation calculations on a series of finite-sized clusters. Each cluster is defined to contain a subset of localized Wannier functions (WFs) (for the occupied space) and projected atomic orbitals (for the virtual space), which can be derived from a periodic Hartree-Fock calculation. Electron-correlation calculations on clusters at second-order Møller-Plesset perturbation theory (MP2) or coupled cluster singles and doubles (CCSD) can be performed with well-established molecular quantum chemistry packages. We perform illustrative calculations at the MP2 and CCSD levels on several types of crystals (neon lattice, carbon monoxide and ammonia crystals, two ionic liquid crystals, and diamond). The results show that CIM is a powerful framework for accurate electron-correlation calculations of crystals.