  Let K be a field with a valuation and let S be the polynomial ring S:=
K[x_1,..., x_n]. We discuss the extension of Groebner theory to ideals in S,
taking the valuations of coefficients into account, and describe the Buchberger
algorithm in this context. In addition we discuss some implementation and
complexity issues. The main motivation comes from tropical geometry, as
tropical varieties can be defined using these Groebner bases, but we also give
examples showing that the resulting Groebner bases can be substantially smaller
than traditional Groebner bases. In the case K = Q with the p-adic valuation
the algorithms have been implemented in a Macaulay 2 package.
