The paper proposes a complex radial basis function network. The network has complex centres and connection weights, but the nonlinearity of its hidden nodes remains a real-valued function. This kind of network is able to generate complicated nonlinear decision surfaced or to approximate an arbitrary nonlinear function in complex multi-dimensional space, and it provides a powerful tool for nonlinear signal processing involving complex signals. The paper is divided into two parts. The first part introduces the network architecture and derives both block-data and recursive learning algorithms for this complex radial basis function network. The complex orthogonal least squares algorithm is a batch learning algorithm capable of constructing an adequate network structure, while a complex version of the hybrid clustering and least squares algorithm offers real-time adaptation capability. The identification of a nonlinear communications channel model is used to illustrate these two learning algorithms. In the second part of the paper, a practical application of this complex radial basis function network is demonstrated using digital communications channel equalisation.