We give a complete characterization of Caratheodory complete pseudoconvex Reinhardt domains, which extends results of Pflug, Fu and the author. It is known that all bounded complete pseudoconvex Reinhardt domains are Caratheodory complete (see [Pfl]). It was S. Fu who extended this result to all bounded Reinhardt pseudoconvex domains satisfying some geometric condition (see [Fu]) indicating that there are domains not satisfying this condition and not being Caratheodory complete. We prove that, actually, each pseudoconvex Reinhardt domain which is Caratheodory complete must fulfill the geometric condition from [Fu]. Moreover, we do not need to restrict ourselves only to bounded domains but we extend the characterization to all Caratheodory hyperbolic pseudoconvex Reinhardt domains, making use of the characterization of such domains from [Zwo], where the characterization of the Kobayashi completeness of the domains under consideration was also given. This altogether gives a complete solution to the problem of characterization of the completeness of pseudoconvex Reinhardt domains with respect to invariant distances. Before we state the main results of the paper let us recall the notation necessary for their formulation. Let D be a domain in C, let E be the unit disk in C and let p denote the Poincare distance on E. For points w, z ∈ D we define cD(w, z) := sup{p(f(w), f(z)) : f ∈ O(D, E)}; kD(w, z) := inf{p(λ1, λ2) : there is f ∈ O(E, D) with f(λ1) = w, f(λ2) = z}; kD := the largest pseudodistance not larger than kD. We call cD (respectively, kD) the Caratheodory (respectively, Kobayashi) pseudodistance. kD is called the Lempert function. Following [Jar-Pfl] we introduce the following notation: dD := tanh dD, where d denotes c, k or k. Received by the editors May 12, 1998. 1991 Mathematics Subject Classification. Primary 32H15; Secondary 32H20, 32F05. The author is a fellow of the Alexander von Humboldt Foundation. c ©1999 American Mathematical Society