We consider one-dimensional models for the irreversible adsorption of large molecules on a solid surface. The study is motivated by recent simulations of the diffusion random sequential adsorption process in which hard spheres diffuse above an adsorbing surface. We first consider a generalized parking process in which the rate of deposition of a particle within a gap formed by two preadsorbed spheres depends on the width of the gap, but is uniform within a gap. We demonstrate simply that all generalized parking processes, including simple random sequential adsorption (RSA), have the same jamming limit coverage. As a by-product of this analysis, we obtain a recursion formula for the saturation coverage in gaps of finite length. In the second part of the paper, we consider a parking process in which the rate of deposition within a gap varies with position as well as the gap width. To apply the model to diffusion random sequential adsorption (DRSA) we solve the steady state diffusion equation to find the probability density function for the creation of a free interval of width h upon adsorption of a particle in a gap of size h'. The resulting jamming limit coverage, θ=0.7506, is in good agreement with the numerical simulations of the DRSA process (0.7496), but larger than that of simple RSA (0.7476). © 1995 The American Physical Society.