Creativity is the ability to produce novel, useful, and surprising ideas, and has been widely studied as a crucial aspect of human cognition. Machine creativity on the other hand has been a long-standing challenge. With the rise of advanced generative AI, there has been renewed interest and debate regarding AI's creative capabilities. Therefore, it is imperative to revisit the state of creativity in AI and identify key progresses and remaining challenges. In this work, we survey leading works studying the creative capabilities of AI systems, focusing on creative problem-solving, linguistic, artistic, and scientific creativity. Our review suggests that while the latest AI models are largely capable of producing linguistically and artistically creative outputs such as poems, images, and musical pieces, they struggle with tasks that require creative problem-solving, abstract thinking and compositionality and their generations suffer from a lack of diversity, originality, long-range incoherence and hallucinations. We also discuss key questions concerning copyright and authorship issues with generative models. Furthermore, we highlight the need for a comprehensive evaluation of creativity that is process-driven and considers several dimensions of creativity. Finally, we propose future research directions to improve the creativity of AI outputs, drawing inspiration from cognitive science and psychology.