Computing Toric Ideals by Lorenzo Robbiano Toric ideals are binomial ideals which represent the algebraic relations of sets of power products. They show up in many fundamental questions arising from several fields of Mathematics, such as Integer Programming, Statistics, Combinatorics, and it is also the basic step of any procedure which computes SAGBI bases. In this introductory exposition I explain how naive approaches to the problem of computing toric ideals may lead to very poor performances, so that the need of new algorithms will be clear.