As soon as the natural polyamines (PAs), putrescine (Put), spermidine (Spd) and spermine (Spm), were recognized as ubiquitous constituents of eukaryotic cells, their involvement in growth-related processes attracted particular interest. The high activities of ornithine decarboxylase (ODC) and S-adenosylmethionine decarboxylase (AdoMetDC) in rapidly growing tissues and cells, particularly in tumour cells, suggested PA biosynthesis as a target for antineoplastic therapy. In the course of the years selective inhibitors have been developed for literally all enzymes of PA metabolism. Some became important as tools in the elucidation of the PA metabolic system, but only few of them were efficient as inhibitors of tumour growth. A major reason for the inefficacy of selective enzyme inhibitors as anticancer drugs is the sophistication of the system, which regulates intracellular PA pools. Selective blockade of a single enzyme induces changes of metabolism and transport, which compensate for the deficit. The selective impairment of tumour growth is in addition hampered by the ubiquitous occurrence of the PAs, their importance in normal functions of nearly all mammalian cells, and the ability of the mammalian organism to utilize exogenous (gastrointestinal) PAs. Among the inhibitors of PA-related enzymes, the ODC inactivator (R, S)-2-(difluoromethyl)ornithine (DFMO) became most famous. Although it was disappointing in most therapeutic attempts to use it as single drug, it has - based on its low toxicity - considerable potential in cancer chemoprevention, and it turned out to be a highly efficient anti-trypanosome agent. Very likely DFMO is suitable to improve the efficacy of some of the current cytotoxic drugs, and it may allow one to create new therapies in combination with other PA-directed drugs. Some of the less selective enzyme inhibitors, particularly those, which inhibit two or more enzymes of PA metabolism, appear to have had a chance to become practically useful, but they have not been developed energetically. Disregarding DFMO, the AdoMetDC inhibitor SAM486A is the only compound for which clinical trials were published. The future of this drug is unclear at present; presumably phase III clinical trials have been discontinued. One of the lessons that had to be learned from the work on selective enzyme inhibitors was that PA metabolism is a much more difficult target, than has been expected on the basis of the simplicity of the PA structures, and the simple reactions involved in their biosynthesis. In order to inhibit tumour growth several reactions or regulatory functions of PA metabolism have to be impaired at the same time. Recent efforts devoted to the development new types of anticancer drugs, which are based on the perturbation of PA metabolism by structural analogues of the natural PAs, take this message into account. These approaches are the topic of the 2nd part of this overview.