This is an excellent article on the problems met in extending the notion of means to positive definite matrices. It should be read by all interested in means. A reasonable set of conditions that a mean \(M\) should satisfy are: (i) \(M(A_1, \dots,M_n)\) is invariant under any permutation of the matrices \(A_i, i\leq i\leq n\); (ii) \(M\) is increasing and continuous in each variable \(A_i, 1\leq i\leq n\); (iii) \(M(X^*A_1X, \dots,X^*A_nX)=X^*M(A_1, \ldots,A_n)X\), \(X\) an invertible matrix. This justifies the well-known but non-intuitive definition of the geometric mean \(G(A,B) = A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}= \sqrt{AB}\) when the matrices commute. Several definitions of \(G(A,B,C)\) lead to an interesting discussion but only one definition appears to satify all of (i)--(iii). The definition of Sagae and Tanabe \[ G(A,B,C)=A^{1/2}\bigl(A^{-1/2}B^{1/2}(B^{-1/2}CB^{-1/2})^{1/2}B^{1/2}A^{-1/2} \bigr)^{2/3} A^{1/2} \] seems to be another possibility that is not discussed.