DeepONets have recently been proposed as a framework for learning nonlinear operators mapping between infinite-dimensional Banach spaces. We analyze DeepONets and prove estimates on the resulting approximation and generalization errors. In particular, we extend the universal approximation property of DeepONets to include measurable mappings in non-compact spaces. By a decomposition of the error into encoding, approximation and reconstruction errors, we prove both lower and upper bounds on the total error, relating it to the spectral decay properties of the covariance operators, associated with the underlying measures. We derive almost optimal error bounds with very general affine reconstructors and with random sensor locations as well as bounds on the generalization error, using covering number arguments. We illustrate our general framework with four prototypical examples of nonlinear operators, namely those arising in a nonlinear forced ordinary differential equation, an elliptic partial differential equation (PDE) with variable coefficients and nonlinear parabolic and hyperbolic PDEs. While the approximation of arbitrary Lipschitz operators by DeepONets to accuracy $\epsilon $ is argued to suffer from a ‘curse of dimensionality’ (requiring a neural networks of exponential size in $1/\epsilon $), in contrast, for all the above concrete examples of interest, we rigorously prove that DeepONets can break this curse of dimensionality (achieving accuracy $\epsilon $ with neural networks of size that can grow algebraically in $1/\epsilon $).Thus, we demonstrate the efficient approximation of a potentially large class of operators with this machine learning framework.