This paper presents a comprehensive survey of event causality identification (ECI), which automatically detects causal relationships between events in texts in the field of natural language processing (NLP). We propose a new classification scheme to systematically categorize and clarify existing methods, and explain the basic principles and technical framework of ECI. We classify ECI methods based on two major tasks: sentence-level event causality identification (SECI) and document-level event causality identification (DECI), and discuss the progress of various methodologies (e.g., feature pattern-based matching, machine learning-based classification, deep semantic encoding, prompt-based fine-tuning, causal knowledge dictionary learning, etc.) and data augmentation strategies for each task, including multilingual and cross-lingual ECI, and zero-shot ECI using large-scale language models (LLMs). We also analyze the strengths and limitations of each method, conduct extensive quantitative evaluations on four benchmark datasets, and suggest future research directions.