Training an artificial neural network with backpropagation algorithms to perform advanced machine learning tasks requires an extensive computational process. This paper proposes to implement the backpropagation algorithm optically for in situ training of both linear and nonlinear diffractive optical neural networks, which enables the acceleration of training speed and improvement in energy efficiency on core computing modules. We demonstrate that the gradient of a loss function with respect to the weights of diffractive layers can be accurately calculated by measuring the forward and backward propagated optical fields based on light reciprocity and phase conjunction principles. The diffractive modulation weights are updated by programming a high-speed spatial light modulator to minimize the error between prediction and target output and perform inference tasks at the speed of light. We numerically validate the effectiveness of our approach on simulated networks for various applications. The proposed in situ optical learning architecture achieves accuracy comparable to in silico training with an electronic computer on the tasks of object classification and matrix-vector multiplication, which further allows the diffractive optical neural network to adapt to system imperfections. Also, the self-adaptive property of our approach facilitates the novel application of the network for all-optical imaging through scattering media. The proposed approach paves the way for robust implementation of large-scale diffractive neural networks to perform distinctive tasks all-optically.