The tropical variety of a polynomial ideal I in n variables over Q is a polyhedral complex in n-dimensional space. We may consider it as a subfan of the Groebner fan of I. The polyhedral cones in the Groebner fan can be computed using Groebner bases and by applying "Groebner walk" techniques. This gives one method for computing the tropical variety of I. We show how the method can be refined by applying a connectivity result for tropical varieties of prime ideals and an algorithm for constructing tropical bases of curves. The presented algorithms have been implemented in the software package Gfan. This is joint work with T. Bogart, K. Fukuda, D. Speyer, B. Sturmfels and R. Thomas.