Substituting rubber particles for a portion of the standard coarse aggregates in concrete is regarded as a sustainable solution for tackling the issue of waste-tires disposal. In order to assess the structural performance of rubber concrete (RC), many studies have been conducted on the proportions, mechanical properties, curing conditions, usages, and serviceability performance of the material over the decades. This review systematically summarizes the mechanical properties (e.g., static and dynamic), testing method, and durability of RC, emphasizing its dynamic characteristics from the perspectives of material and component. The inclusion of rubber particles weakens the static properties of the concrete, while the low module of inherent rubbers improves the concrete dynamic properties, such as low stiffness degradation, high strain-rate sensitivity, excellent energy dissipations, and good ductility. With the increase in the strain rate, the improvement in energy absorption and ductility of the RC (0 to 30%) can increase to 110% and 80%, respectively. Concrete with a rubber volume fraction of less than 30% enhances both mechanical and long-term environmental performances. Moreover, RC shows good fire resistance, permeability, and freeze–thaw behavior; however, further research is needed to understand its constitutive model and the synergistic effects of additional materials.