Complex

$n\times n$ matrices A, B are said to be consimilar if

$B={S}^{-1}A\overline{S}$ for some non-singular complex matrix S. This concept arises naturally from comparing the expressions for a semilinear transformation on an n-dimensional complex vector space in two different coordinate systems. The present paper gives a detailed survey, with an extensive bibliography, of the known results on consimilarity. A canonical form for A under consimilarity, closely related to the usual Jordan normal form for

$A\overline{A}$, is described in section 3. Various applications are given in section 4. For example (Theorem 4.5), A and B are consimilar if, and only if, (a)

$A\overline{A}$ and

$B\overline{B}$ are similar and (b)

$A,A\overline{A},A\overline{A}A,\xb7\xb7\xb7$ have the same respective ranks as

$B,B\overline{B},B\overline{B}B,\xb7\xb7\xb7$. Again, it is noted that A is consimilar to

$\overline{A},$ ${A}^{\top}$ and

${A}^{*}$, and that every matrix is consimilar both to a real matrix and to a Hermitian matrix. Altogether, this is a useful and clearly written survey.