Massively parallel simulations based on discontinuous-Galerkin (DG) PDE solvers often show poor scalability at extreme scales. This is mainly attributed to data communication and synchronization between different processing elements (PEs). This paper presents an asynchronous DG method that relaxes communication/synchronization at a mathematical level and allows computations at PEs to proceed regardless of the communication status, thus increasing the computation-communication overlap. We show that standard DG schemes under relaxed synchronization result in a loss of conservation property and provide solutions that are at most first-order accurate. Subsequently, we describe a simple method to preserve conservation and develop new asynchrony-tolerant (AT) fluxes that provide high-order accurate solutions. Results from simulations of one-dimensional linear and nonlinear equations are presented to verify the accuracy of the asynchronous DG method.