Acoustic and elastic waves carry a wealth of useful physical information in real world. Sensing acoustic and elastic waves is very important for discovering knowledge in various fields. Conventional wave sensing approaches generally require multiple expensive sensors and complex hardware systems due to the uniform spatial transmission characteristics of physical fields. These limitations prompt the development of wave sensing strategies with high integration degree, lightweight structure, and low hardware cost. Due to their extraordinary physical properties, artificially engineered structures such as metastructures can encode the physical field information by flexibly manipulating the transmission characteristics of acoustic and elastic waves. The fusion of information coding and wave sensing process breaks through the limitations of conventional sensing approaches and reduces the sensing cost. This review aims to introduce the advances in spatial information coding with artificially engineered structures for acoustic and elastic wave sensing. First, we review the enhanced spatial wave sensing with metastructures for weak signal detection and source localization. Second, we introduce computational sensing approaches that combines the spatial transmission coding structures with reconstruction algorithms. Representative progress of computational sensing with metastructures and random scattering media in audio source separation, ultrasonic imaging, and vibration information identification is reviewed. Finally, the open problems, challenges, and research prospects of the spatial information coding structures for acoustic and elastic wave sensing are discussed.