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Music is a powerful medium for altering the emotional state of the listener. In recent years, with significant advancement in computing capabilities, artificial intelligence-based (AI-based) approaches have become popular for creating affective music generation (AMG) systems that are empowered with the ability to generate affective music. Entertainment, healthcare, and sensor-integrated interactive system design are a few of the areas in which AI-based affective music generation (AI-AMG) systems may have a significant impact. Given the surge of interest in this topic, this article aims to provide a comprehensive review of AI-AMG systems. The main building blocks of an AI-AMG system are discussed, and existing systems are formally categorized based on the core algorithm used for music generation. In addition, this article discusses the main musical features employed to compose affective music, along with the respective AI-based approaches used for tailoring them. Lastly, the main challenges and open questions in this field, as well as their potential solutions, are presented to guide future research. We hope that this review will be useful for readers seeking to understand the state-of-the-art in AI-AMG systems, and gain an overview of the methods used for developing them, thereby helping them explore this field in the future.