We propose a generalized, lattice-based statistical thermodynamic theory for understanding the interfacial phenomena in systems containing hydrogen bonding molecules (often termed as associating molecules), such as water : amphiphiles, block copolymers and associating solid surfaces. The basic assumption is that the configurational partition function (Q) can be factored into two parts : (i) one term [Q((phys))] arising from the presence of nonassociating, or the "physical," interactions, for which we adopt the self-consistent-field theory [Scheutjens and Fleer, J. Phys. Chem. 84, 178 (1980)], (ii) the other term [Q((hbond))] arising from the presence of hydrogen bond interactions, for which we propose a new association theory. The focus of the proposed association theory is on the correct counting of the number of II bonds that are formed between various types of donor and acceptor sites that satisfy the proximity and orientational requirements for bond formation. The expression for Q((hbond)) is evaluated by accounting for the entropic loss and energy released upon the formation of each hydrogen bond, and the transient nature of hydrogen bonds. The equilibrium criteria for H bonding is satisfied by minimizing the free energy of the system with respect to the number of H bonds formed between each type of donor site present in each layer z and each type of acceptor site present in each layer z＇, where z＇ = z, or z +/- 1. It turns out that the final expression for Q((hbond)) , at equilibrium, depends only on the fraction of unbonded association sites of all types that are located at various distances from the interface, which are themselves related to the equilibrium constant of formation of H bond between various donor-acceptor pairs, temperature of the fluid and the concentration profile in the interfacial region. For systems containing pure, spherical, associating molecules in the fluid phase, our expression for Q((hbond)) is found to;be identical to that of the density functional theory [Segura et al., Mel. Phys. 90, 759 (1997)], except for the inherent differences existing between continuum and lattice treatments. We present the results of the proposed theory in two parts. First, we verify the thermodynamic consistency of our approach with the Gibbs adsorption rule. Second, to clearly elucidate the role of hydrogen bonding on interfacial properties, we provide results for systems containing a binary fluid mixture, which comprises of an associating monomeric solvent and an amphiphilic, di-block, chain molecule, against an associating solid surface.