Despite being responsible for state-of-the-art results in several computer vision and natural language processing tasks, neural networks have faced harsh criticism due to some of their current shortcomings. One of them is that neural networks are correlation machines prone to model biases within the data instead of focusing on actual useful causal relationships. This problem is particularly serious in application domains affected by aspects such as race, gender, and age. To prevent models from incurring on unfair decision-making, the AI community has concentrated efforts in correcting algorithmic biases, giving rise to the research area now widely known as fairness in AI. In this survey paper, we provide an in-depth overview of the main debiasing methods for fairness-aware neural networks in the context of vision and language research. We propose a novel taxonomy to better organize the literature on debiasing methods for fairness, and we discuss the current challenges, trends, and important future work directions for the interested researcher and practitioner.