The pathological characteristics of tumour cells often include variation of their histopathological features (i.e. "degrees of de-differentiation") between cases of the same tumour type and between different foci within individual tumours. Usually, only a few cell lines from tumours are immortal. Currently, somatic mutation, replicative infidelity of DNA and aneuploidy are suggested as alternative mechanisms of genomic disturbance underlying tumours. Nevertheless, apart from Hansemann's ideas of "anaplasia" and "de-differentiation" (proposed in the 1890s), and supposed "evolutionary themes" in cancer cell biology, little has been published concerning how histopathologic variation and immortality in tumour cells might arise. This paper reviews applications of the concepts of "variation" to tumours, including concepts of "evolution" and "cellular Darwinism". It is proposed that combinations of somatic mutation, DNA replicative infidelity and aneuploidy may explain the variabilities in tumours, and provide immortality in occasional tumour cells. A possible model involves (i) an initial somatic mutation causing reduced replicative fidelity of DNA, which could be variable in intensity, and thus give rise to variations between cases; (ii) a phase of replicative infidelity of DNA causing daughter cells lines to develop various abnormalities to different degrees, and hence provide for variation between areas of the same tumour. As a last event (iii) occasional asymmetric chromosomal distributions (aneuploidy) might "refresh" the ability of a daughter cell to replicate DNA faithfully causing them to become immortal. Thus extensively mutant and variable, hyperploid, and occasionally immortal cells might arise.