Despite recent advances in modern machine learning algorithms, the opaqueness of their underlying mechanisms continues to be an obstacle in adoption. To instill confidence and trust in artificial intelligence (AI) systems, explainable AI (XAI) has emerged as a response to improve modern machine learning algorithms' explainability. Inductive logic programming (ILP), a subfield of symbolic AI, plays a promising role in generating interpretable explanations because of its intuitive logic-driven framework. ILP effectively leverages abductive reasoning to generate explainable first-order clausal theories from examples and background knowledge. However, several challenges in developing methods inspired by ILP need to be addressed for their successful application in practice. For example, the existing ILP systems often have a vast solution space, and the induced solutions are very sensitive to noises and disturbances. This survey paper summarizes the recent advances in ILP and a discussion of statistical relational learning (SRL) and neural-symbolic algorithms, which offer synergistic views to ILP. Following a critical review of the recent advances, we delineate observed challenges and highlight potential avenues of further ILP-motivated research toward developing self-explanatory AI systems.