The article is devoted to revealing further kinds of the meaning of the term 'efficiency' (and related notions), consequently setting them in mathematical frames. First of all the so called envelope-type efficiency is introduced. This notion is illustrated by several examples derived from the elementary topology, Bayesian statistics, mathematical economics and primer of financial engineering. It seems that the above examples do reflect the essence of this idea in the best possible picture. The next proposition concerns the type of efficiency which was called 'collective-type efficiency'. It turns out that the reasonable compromise is 'better than the best solution' (even in the Nash sense). The quite good class of examples are provided by problems derived from the famous 'prisoner's dilemma' and exploitation of common resources. At the end of the paper some complementing thoughts are indicated - in a loose form. They concern the principal conflict between efficiency and equity as well as the problem of economic behaviour in the sphere of scientific research (the balance between two factors: 'erudite components' and 'creative potentials' of the researchers).