Bioinspired engineering based on biosonar systems in nature is reviewed and discussed in terms of the merits of different approaches and their results: biosonar systems are attractive technological paragons because of their capabilities, built-in task-specific knowledge, intelligent system integration and diversity. Insights from the diverse set of sensing tasks solved by bats are relevant to a wide range of application areas such as sonar, biomedical ultrasound, non-destructive testing, sensors for autonomous systems and wireless communication. Challenges in the design of bioinspired sonar systems are posed by transducer performance, actuation for sensor mobility, design, actuation and integration of beamforming baffle shapes, echo encoding for signal processing, estimation algorithms and their implementations, as well as system integration and feedback control. The discussed examples of experimental systems have capabilities that include localization and tracking using binaural and multiple-band hearing as well as self-generated dynamic cues, classification of small deterministic and large random targets, beamforming with bioinspired baffle shapes, neuromorphic spike processing, artifact rejection in sonar maps and passing range estimation. In future research, bioinspired engineering could capitalize on some of its strengths to serve as a model system for basic automation methodologies for the bioinspired engineering process.