Given the tremendous potential and influence of artificial intelligence (AI) and algorithmic decision-making (DM), these systems have found wide-ranging applications across diverse fields, including education, business, healthcare industries, government, and justice sectors. While AI and DM offer significant benefits, they also carry the risk of unfavourable outcomes for users and society. As a result, ensuring the safety, reliability, and trustworthiness of these systems becomes crucial. This article aims to provide a comprehensive review of the synergy between AI and DM, focussing on the importance of trustworthiness. The review addresses the following four key questions, guiding readers towards a deeper understanding of this topic: (i) why do we need trustworthy AI? (ii) what are the requirements for trustworthy AI? In line with this second question, the key requirements that establish the trustworthiness of these systems have been explained, including explainability, accountability, robustness, fairness, acceptance of AI, privacy, accuracy, reproducibility, and human agency, and oversight. (iii) how can we have trustworthy data? and (iv) what are the priorities in terms of trustworthy requirements for challenging applications? Regarding this last question, six different applications have been discussed, including trustworthy AI in education, environmental science, 5G-based IoT networks, robotics for architecture, engineering and construction, financial technology, and healthcare. The review emphasises the need to address trustworthiness in AI systems before their deployment in order to achieve the AI goal for good. An example is provided that demonstrates how trustworthy AI can be employed to eliminate bias in human resources management systems. The insights and recommendations presented in this paper will serve as a valuable guide for AI researchers seeking to achieve trustworthiness in their applications.