It is widely known that neural networks (NNs) are universal approximators of functions. However, a less known but powerful result is that a NN can accurately approximate any nonlinear operator. This universal approximation theorem of operators is suggestive of the potential of deep neural networks (DNNs) in learning operators of complex systems. In this talk, I will present the deep operator network (DeepONet) to learn various operators that represent deterministic and stochastic differential equations. I will also present several extensions of DeepONet, such as DeepM&Mnet for multiphysics problems, DeepONet with proper orthogonal decomposition (POD-DeepONet), (Fourier-)MIONet for multiple-input operators, and multifidelity DeepONet. I will demonstrate the effectiveness of DeepONet and its extensions to diverse multiphysics and multiscale problems, such as nanoscale heat transport, bubble growth dynamics, high-speed boundary layers, electroconvection, hypersonics, and geological carbon sequestration. Deep learning models are usually limited to interpolation scenarios, and I will quantify the extrapolation complexity and develop a complete workflow to address the challenge of extrapolation for deep neural operators.