The concepts of diversity and biodiversity are analysed regarding their historical emergence, and their intrinsic meaning and differences are discussed. Through a brief synopsis, difficulties usually experienced by statisticians in capturing the dynamics of diversity are analysed and main problems identified. The shift from diversity to the more holistic biodiversity as a working concept is appraised in terms of the novelty involved. Through a number of examples, the way the two concepts capture natural cyclic changes is analysed, and their reciprocal and complementary relations are approached theoretically. The way diversity could develop from the stores of biodiversity as its active expression through selective and evolutionary processes is described. Through the use of a very simple dynamic model, the concepts of diversity and biodiversity are analysed in extremely opposite hypothetical scenarios. Comparisons with natural situations are made and the theoretical implications from the conservation point of view are discussed. These support the opinion that conservation undertaken in restricted and protected areas is not self-sustainable, needing permanent external intervention to regulate internal processes, and in the long run will most probably lead in the direction of obsolescence and extinction. Finally, the relations between diversity, biodiversity, and sustainability are approached. The vagueness of the sustainability concept is discussed. Preservation of biodiversity is then defended as one of the best available indicators to assist us in fixing boundaries which may help to provide a more precise definition of sustainability.