In the paper we consider distributed algorithms for greedy graph coloring. For the largest-first (LF) approach, we propose a new distributed algorithm which is shown to color a graph in an expected time of O(ΔlognlogΔ) rounds, and we prove that any distributed LF-coloring algorithm requires at least Ω(Δ) rounds. We discuss the quality of obtained colorings in the general case and for particular graph classes. Finally, we show that other greedy graph coloring approaches, such as smallest-last (SL) or dynamic-saturation (SLF), are not suitable for application in distributed computing, requiring Ω(n) rounds.