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Artificial intelligence (AI) systems utilizing deep neural networks (DNNs) and machine learning (ML) algorithms are widely used for solving important problems in bioinformatics, biomedical informatics, and precision medicine. However, complex DNNs or ML models, which are often perceived as opaque and black-box, can make it difficult to understand the reasoning behind their decisions. This lack of transparency can be a challenge for both end-users and decision-makers, as well as AI developers. Additionally, in sensitive areas like healthcare, explainability and accountability are not only desirable but also legally required for AI systems that can have a significant impact on human lives. Fairness is another growing concern, as algorithmic decisions should not show bias or discrimination towards certain groups or individuals based on sensitive attributes. Explainable artificial intelligence (XAI) aims to overcome the opaqueness of black-box models and provide transparency in how AI systems make decisions. Interpretable ML models can explain how they make predictions and the factors that influence their outcomes. However, most state-of-the-art interpretable ML methods are domain-agnostic and evolved from fields like computer vision, automated reasoning, or statistics, making direct application to bioinformatics problems challenging without customization and domain-specific adaptation. In this paper, we discuss the importance of explainability in the context of bioinformatics, provide an overview of model-specific and model-agnostic interpretable ML methods and tools, and outline their potential caveats and drawbacks. Besides, we discuss how to customize existing interpretable ML methods for bioinformatics problems. Nevertheless, we demonstrate how XAI methods can improve transparency through case studies in bioimaging, cancer genomics, and text mining.