In this paper we present a novel method, called Evolutionary Acquisition of Neural Topologies (EANT), of evolving the structure and weights of neural networks. The method introduces an efficient and compact genetic encoding of a neural network onto a linear genome that enables one to evaluate the network without decoding it. The method explores new structures whenever it is not possible to further exploit the structures found so far. This enables it to find minimal neural structures for solving a given learning task. We tested the algorithm on a benchmark control task and found it to perform very well.