Canonical Nambu mechanics: Relevance to string/M-theory and approaches to quantization

We review some aspects of Nambu mechanics on the basis of works previously published separately by the present author. The main focuses are on three themes: the various symmetry structures, their possible relevance to string/M-theory, and a Hamilton-Jacobi-like reformulation. We try to elucidate the basic ideas, most of which were rooted in more or less the same ground, and to explain the motivations behind these works from a unified and vantage viewpoint. Various unsolved questions are mentioned. We also include a historical account of the genesis of Nambu mechanics, and discuss (in the appendix) some parallelism of various ideas behind Nambu's paper with Dirac's old works which are related to the description of vortical flows in terms of gauge potentials.