All-solid-state batteries, employing inorganic ion conductors as electrolytes, can surpass the current Li-ion technology in terms of energy density, battery safety, specific power, as well as a fast-charging capability; however, a highly conductive solid electrolyte is essential. While recent extensive explorations of solid ion conductors have led to a list of candidate materials, there are still enormous variations of the ionic conductivity even within the same class of the materials, indicating the strong influence of structural modifications on the ion transport. In this review, besides revisiting general strategies of materials design for fast ion transport, we summarize the present state of affairs of promising classes of crystalline solid electrolytes, including a structural description and an overview of the observed static lattice effects alongside with open questions specific for the pertinent material classes. In the end, future directions and open questions to design and develop solid electrolytes, i.e. upcoming classes of materials, influence of lattice dynamics and inductive effects, the origin of energy landscape flattening, and the impact of synthesis routes are discussed. We hope this review provides a shape of the current status of the field of crystalline ion conductors.