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The problem of the complexity of Buchberger's algorithm to compute GrSbner bases has been recently studied by Buchberger [2,3],Baye~ l],and Lazar~ 7].Here we present some results on this question,by giving both lower and upper bounds for the maximal degree of the elements of a Gr6bner basis of a polynomial ideal,as a function of the degree of a general basis,the number of variables and the dimension of the ideal.To know the complexity of Buchberger's algorithm,the knowledge of a bound for the degree of the GrSbner basis is not sufficient:also a bound for the degree of the polynomials arising during the intermediate computations is required. Unfortunately we are able to obtain such a bound only under assumptions on the input basis (namely,an Hbasis of the ideal must be known). Our examples show that this condition is unavoidable.ii) the second class of examples is more related with a kind of basis (H-basis) which has weaker properties than a Gr6bner basis and which is widely studied in classical constructive ideal theory.ln her well-known pape~ 61 ,Hermann gave an incorrect proof that the maximal degree of an H-basis is doubly exponential in the number of variables.Here we adapt an example by Mayr and Meyer[8] to show the existence of ideals such that any H-basis (and henceforth any Gr~bner basis under a degree compatible term-ordering)contains elements whose degree is doubly exponential in the dimension of the ideal.As for upper bounds,Bayer [l] proved that the degree of a Gr6bner basis is bounded by a coefficient (under a suitable representation) of the Hilbert polynomial of the ideal (the use of Hilbert polynomials to give a bound to the degree of a Gr6bner basis has