Finite order variational sequences: a short review Raffaele Vitolo Department of Mathematics “E. De Giorgi”, University of Lecce Via per Arnesano, 73100 Lecce, Italy E–mail: raffaele.vitolo@unile.it Web: http://poincare.unile.it/vitolo Dedicated to Demeter Krupka in honour of his 65th birthday Proceedings of the Colloquium on Variations, Geometry and Physics in honour of Demeter Krupka’s 65th birthday Olomouc, 25-26 August 2007 Nova Science Publisher (2008), 117–136. Abstract Variational sequences are complexes of modules or sheaf sequences in which one of the maps is the Euler–Lagrange operator, i.e., the differential operator taking a Lagrangian into its Euler–Lagrange form. In this review paper we dis- cuss variational sequences on finite order jets, with special emphasis on Krupka’s approach. We also discuss recent results on this topic as well as possible research directions. Key words: Jet spaces, variational sequence, variational bicomplex. 2000 MSC: primary 58J10, secondary 58A12, 58A20. 1