This review paper explores existing architectures, operating principles, performance metrics and applications of analog Gaussian function circuits. Architectures based on the translinear principle, the bulk-controlled approach, the floating gate approach, the use of multiple differential pairs, compositions of different fundamental blocks and others are considered. Applications involving analog implementations of Machine Learning algorithms, neuromorphic circuits, smart sensor systems and fuzzy/neuro-fuzzy systems are discussed, focusing on the role of the Gaussian function circuit. Finally, a general discussion and concluding remarks are provided.