We discuss some general properties of the wave functions of H-2(+) in two dimensions, when the electron is close to the nucleus, and when it is far away from the nucleus. Based on these properties, some simple, parameter-free wave functions are developed for some of the states. They lead to quite accurate energies and give an insight into the structure of H-2(+) in two dimensions. These wave functions and appropriate correlation functions, are then used to develop reliable wave functions for H-2 in two dimensions. They provide a clear and interesting comparison with the corresponding results in three dimensions. (C) 2003 American Institute of Physics.