Respiratory diseases represent a significant global burden, necessitating efficient diagnostic methods for timely intervention. Digital biomarkers based on audio, acoustics, and sound from the upper and lower respiratory system, as well as the voice, have emerged as valuable indicators of respiratory functionality. Recent advancements in machine learning (ML) algorithms offer promising avenues for the identification and diagnosis of respiratory diseases through the analysis and processing of such audio-based biomarkers. An ever-increasing number of studies employ ML techniques to extract meaningful information from audio biomarkers. Beyond disease identification, these studies explore diverse aspects such as the recognition of cough sounds amidst environmental noise, the analysis of respiratory sounds to detect respiratory symptoms like wheezes and crackles, as well as the analysis of the voice/speech for the evaluation of human voice abnormalities. To provide a more in-depth analysis, this review examines 75 relevant audio analysis studies across three distinct areas of concern based on respiratory diseases' symptoms: (a) cough detection, (b) lower respiratory symptoms identification, and (c) diagnostics from the voice and speech. Furthermore, publicly available datasets commonly utilized in this domain are presented. It is observed that research trends are influenced by the pandemic, with a surge in studies on COVID-19 diagnosis, mobile data acquisition, and remote diagnosis systems.