The multi-layer structures of Deep Learning facilitate the processing of higher-level abstractions from data, thus leading to improved generalization and widespread applications in diverse domains with various types of data. Each domain and data type presents its own set of challenges. Real-world time series data may have a non-stationary data distribution that may lead to Deep Learning models facing the problem of catastrophic forgetting, with the abrupt loss of previously learned knowledge. Continual learning is a paradigm of machine learning to handle situations when the stationarity of the datasets may no longer be true or required. This paper presents a systematic review of the recent Deep Learning applications of sensor time series, the need for advanced preprocessing techniques for some sensor environments, as well as the summaries of how to deploy Deep Learning in time series modeling while alleviating catastrophic forgetting with continual learning methods. The selected case studies cover a wide collection of various sensor time series applications and can illustrate how to deploy tailor-made Deep Learning, advanced preprocessing techniques, and continual learning algorithms from practical, real-world application aspects.