A theoretical study of the highest bound states and some of the lowest energy resonance states of water in its ground electronic state, (X) over tilde = 1(1)A ', and with zero total angular momentum, is presented. The majority of our calculations correspond to the even symmetry block with respect to hydrogen atom exchange. An accurate ab initio based potential surface is employed. The Lanczos method, applied to a large grid representation of the Hamiltonian, is shown to be a straightforward method for obtaining the bound state energy levels. Selected eigenfunctions are also determined and several of those near the dissociation threshold are quite extended. Resonance states just above the dissociation threshold are characterized with the aid of damped Chebyshev iterations. Among the resonances observed are those with hyperspherical and local mode character.