  All real three dimensional Poisson-Lie groups are explicitly constructed and
fully classified under group automorphisms by making use of their one-to-one
correspondence with the complete classification of real three-dimensional Lie
bialgebras given in [X. Gomez, J. Math. Phys. vol. 41, p. 4939 (2000)]. Many of
these 3D Poisson-Lie groups are non-coboundary structures, whose Poisson
brackets are given here for the first time. Casimir functions for all
three-dimensional PL groups are given, and some features of several PL
structures are commented.
