<p>In this paper, we extend the compact central schemes for hyperbolic conservation laws to solve a class of conservative partial differential equations (PDEs) with arbitrarily high-order spatial-temporal accuracy. The proposed schemes inherit most advantages of the original compact central schemes, namely possessing highly compact spatial stencils and achieving high-order time accuracy with one step. Moreover, the proposed schemes with arbitrarily high-order in both space and time can be expressed by uniform formulae without additional calculations and derivations. Therefore, we can easily implement the proposed schemes with an arbitrary order in a single and simple computer program. The linear stabilities of the proposed schemes for different types of PDEs are analyzed. Numerical experiments for several kinds of classical PDEs are provided to demonstrate the accuracy of the numerical methods.</p>