This paper is based on an expository talk given at the International Conference on Radicals and Rings ICOR 97. Our aim is not to give a comprehensive survey of the subject. We rather wish to present some selected results and on their basis to analyze the development of radical theory of near-rings, emphasizing the role of non-zerosymmetric near-rings. Besides we present some new results on Jacobson type radicals of non-zerosymmetric near-rings.