The sigmoid function is a mathematical function that produces an S-shaped curve, often used in statistics and machine learning. It takes any real-valued number and maps it to a value between 0 and 1. This property makes it particularly useful for modeling probabilities, as it can represent the likelihood of a binary outcome. In the context of neural networks, the sigmoid function serves as an activation function, helping to determine whether a neuron should be activated based on its input. Its smooth gradient allows for effective optimization during the training process, although it has been largely replaced by other functions like ReLU in many applications due to issues like vanishing gradients.