The paper is an exposition of some possible generalization of the Hirzebruch-Riemann-Roch theorem: \(\chi({\mathcal E})=\deg (ch({\mathcal E})\cdot td(T_ X))_ n\) where \({\mathcal E}\) is a locally free sheaf of a smooth projective variety of dimension n over an algebraically closed field. Grothendieck’s formulation of the R-R-theorem casts lights upon possible extensions to a more general setting, i.e. to larger categories of schemes and relative morphisms. The first problem is the definition of a functor A from this category to rings (which will play the role of the classical Chow ring) and a natural transformation from the Grothendieck K-functor to A (which will replace the Chern character). For instance, in the case of the category of quasi-projective (possible singular) varieties Baum, Fulton and McPherson have defined Chow groups and intersection theory which give a R-R-theorem for proper morphism. This has been extended to arbitrary algebraic schemes by Fulton and Gillet. Another possible approach is to introduce a suitable filtration on \(K_ 0(X)\) and take \(A(X)\) to be the graded ring associated to this filtration (after tensoring with \({\mathbb{Q}})\). The author reviews the \(\lambda\)-ring structure on \(K_ 0(X)\) and the associated \(\chi\)-filtration and puts it in relation with the “topological” filtration and the Fulton-Lang filtration presenting some new results, too. Using the Quillen filtration on \(K_ 0(X)\) defined by cohomology groups, the author presents Bloch’s formula for quasi projective non-singular varieties proved by Quillen and for varieties with isolated singularities proved by the author and Weibel. The paper contains some examples and open questions.