In the summer of 2005, a meeting of the Consultative Committee for Units of the International Committee on Weights and Measures took place. One of the topics discussed at the meeting was a possible redefinition of the kilogram in terms of fundamental physical constants — a question of relevance to a wide circle of specialists, from school teachers to physicists performing research in a great variety of fields. In this paper, the current situation regarding this question is briefly reviewed and its discussion at the Consultative Committee for Units and other bodies involved is covered. Other issues related to the International System of Units (SI) and broached at the meeting are also discussed.