Astronomers have long learned to classify spirals in terms of their morphological features, primarily their bulges, discs, and bars. In addition other properties of discs are used in classification, notably spiral arms, warps and truncations. In this paper we show how, with modern techniques, parts of the basis of these standard classfications are shown to be unsatisfactory, even misleading. In particular we will look carefully at bulges and truncations, and explain how on purely morphological grounds we need to make basic changes in our concepts. We will put in question the traditional view of the spheroidal bulge, and will also show that the majority of galaxy discs ar not in fact truncated.