The application of mechanics to biology--biomechanics--bears great challenges due to the intricacy of living things. Their dynamism, along with the complexity of their mechanical response (which in itself involves complex chemical, electrical, and thermal phenomena) makes it very difficult to correlate empirical data with theoretical models. This difficulty elevates the importance of useful biomechanical theories compared to other fields of engineering. Despite inherent imperfections of all theories, a well formulated theory is crucial in any field of science because it is the basis for interpreting observations. This is all-the-more vital, for instance, when diagnosing symptoms, or planning treatment to a disease. The notion of interpreting empirical data without theory is unscientific and unsound. This paper attempts to fortify the importance of biomechanics and invigorate research efforts for those engineers and mechanicians who are not yet involved in the field. It is not aimed here, however, to give an overview of biomechanics. Instead, three unsolved problems are formulated to challenge the readers. At the micro-scale, the problem of the structural organization and integrity of the living cell is presented. At the meso-scale, the enigma of fingerprint formation is discussed. At the macro-scale, the problem of predicting aneurysm ruptures is reviewed. It is aimed here to attract the attention of engineers and mechanicians to problems in biomechanics which, in the author's opinion, will dominate the development of engineering and mechanics in forthcoming years.