Speech technology is not appropriately explored even though modern advances in speech technology-especially those driven by deep learning (DL) technology-offer unprecedented opportunities for transforming the healthcare industry. In this paper, we have focused on the enormous potential of speech technology for revolutionising the healthcare domain. More specifically, we review the state-of-the-art approaches in automatic speech recognition (ASR), speech synthesis or text to speech (TTS), and health detection and monitoring using speech signals. We also present a comprehensive overview of various challenges hindering the growth of speech-based services in healthcare. To make speech-based healthcare solutions more prevalent, we discuss open issues and suggest some possible research directions aimed at fully leveraging the advantages of other technologies for making speech-based healthcare solutions more effective.