Sleep is crucial for human health from metabolic, mental, emotional, and social points of view; obtaining good sleep in terms of quality and duration is fundamental for maintaining a good life quality. Over the years, several systems have been proposed in the scientific literature and on the market to derive metrics used to quantify sleep quality as well as detect sleep disturbances and disorders. In this field, wearable systems have an important role in the discreet, accurate, and long-term detection of biophysical markers useful to determine sleep quality. This paper presents the current state-of-the-art wearable systems and software tools for sleep staging and detecting sleep disorders and dysfunctions. At first, the paper discusses sleep’s functions and the importance of monitoring sleep to detect eventual sleep disturbance and disorders. Afterward, an overview of prototype and commercial headband-like wearable devices to monitor sleep is presented, both reported in the scientific literature and on the market, allowing unobtrusive and accurate detection of sleep quality markers. Furthermore, a survey of scientific works related the effect of the COVID-19 pandemic on sleep functions, attributable to both infection and lifestyle changes. In addition, a survey of algorithms for sleep staging and detecting sleep disorders is introduced based on an analysis of single or multiple biosignals (EEG—electroencephalography, ECG—electrocardiography, EMG—electromyography, EOG—electrooculography, etc.). Lastly, comparative analyses and insights are provided to determine the future trends related to sleep monitoring systems.