To handle real-world complexities, intelligent systems need to incrementally acquire, update, and use knowledge throughout their lifetime, a capability known as continual learning (CL). However, neural network training processes face the challenge of catastrophic forgetting (CF), where learning new tasks degrades performance on previously learned ones. This survey provides a comprehensive overview of CL, including fundamental concepts, theoretical frameworks, methodologies, and practical implementations. Through empirical analysis and benchmarking, it highlights the strengths and weaknesses of state-of-the-art CL methods. This paper also provides an overview of Machine Unlearning (MU), an emerging paradigm that removes previously learned training data from a trained model, including fundamental concepts, methodologies, and its connections to CL. It also provides a mathematical analysis examining the effect of CF on MU, identifying it as one of the key research directions for facilitating the process of lifelong deep learning in dynamic environments.