Six different canonical forms for the tripleA, B, Cof a linear multivariable system are discussed, namely (i) the column-companion form, (ii) the row-companion form, (iii) the output identifiable form, (iv) the input-identifiable form, (v) the controllable canonical form and (vi) the observable canonical form. Simple algorithms for transforming any system into these forms are proposed and illustrated by means of examples. Practical applications of the various forms are discussed.