As the range of decisions made by Artificial Intelligence (AI) expands, the need for Explainable AI (XAI) becomes increasingly critical. The reasoning behind the specific outcomes of complex and opaque financial models requires a thorough justification to improve risk assessment, minimise the loss of trust, and promote a more resilient and trustworthy financial ecosystem. This Systematic Literature Review (SLR) identifies 138 relevant articles from 2005 to 2022 and highlights empirical examples demonstrating XAI's potential benefits in the financial industry. We classified the articles according to the financial tasks addressed by AI using XAI, the variation in XAI methods between applications and tasks, and the development and application of new XAI methods. The most popular financial tasks addressed by the AI using XAI were credit management, stock price predictions, and fraud detection. The three most commonly employed AI black-box techniques in finance whose explainability was evaluated were Artificial Neural Networks (ANN), Extreme Gradient Boosting (XGBoost), and Random Forest. Most of the examined publications utilise feature importance, Shapley additive explanations (SHAP), and rule-based methods. In addition, they employ explainability frameworks that integrate multiple XAI techniques. We also concisely define the existing challenges, requirements, and unresolved issues in applying XAI in the financial sector.