Animal pose estimation (APE) aims to locate the animal body parts using a diverse array of sensor and modality inputs (e.g. RGB cameras, LiDAR, infrared, IMU, acoustic and language cues), which is crucial for research across neuroscience, biomechanics, and veterinary medicine. By evaluating 176 papers since 2011, APE methods are categorised by their input sensor and modality types, output forms, learning paradigms, experimental setup, and application domains, presenting detailed analyses of current trends, challenges, and future directions in single- and multi-modality APE systems. The analysis also highlights the transition between human and animal pose estimation, and how innovations in APE can reciprocally enrich human pose estimation and the broader machine learning paradigm. Additionally, 2D and 3D APE datasets and evaluation metrics based on different sensors and modalities are provided. A regularly updated project page is provided here: https://github.com/ChennyDeng/MM-APE.