The paper proposes an articial neural network (ANN) being a global approximator for a special class of functions, which are known as generalized homogeneous. The homogeneity means a symmetry of a function with respect to a group of transformations having topological characterization of a dilation. In this paper, a class of the so-called linear dilations is considered. A homogeneous universal approximation theorem is proven. Procedures for a transformation of an existing ANN to a homogeneous one are developed. Theoretical results are illustrated by several academic examples highlighting potential applications of homogeneous ANNs in various domains such as systems theory, automatic control and computer science. In particular, a scaling invariant patter recognition by homogeneous ANN is demonstrated. An ANN-based identication of generalized homogeneous dynamical system is considered. A feedback for robust homogeneous stabilization of multi-input linear time-invariant system is designed using the proposed ANN.