Abstract. We study irreversible polymer adsorption from dilute solutions theoretically. Universal features of the resultant non-equilibrium layers are predicted. Two broad cases are considered, distinguished by the magnitude of the local monomer-surface sticking rate Q: chemisorption (very small Q) and physisorption (large Q). Early stages of layer formation entail single chain adsorption. While single chain physisorption times τ ads are typically micro to milli-seconds, for chemisorbing chains of N units we find experimentally accessible times τ ads = Q −1 N 3/5 , ranging from seconds to hours. We establish 3 chemisorption universality classes, determined by a critical contact exponent: zipping, accelerated zipping and homogeneous collapse. For dilute solutions, the mechanism is accelerated zipping: zipping propagates outwards from the first attachment, accelerated by occasional formation of large loops which nucleate further zipping. This leads to a transient distribution ω(s) ∼ s −7/5 of loop lengths s up to a maximum size s max ≈ (Qt) 5/3 after time t. By times of order τ ads the entire chain is adsorbed. The outcome of the single chain adsorption episode is a monolayer of fully collapsed chains. Having only a few vacant sites to adsorb onto, late arriving chains form a diffuse outer layer. In a simple picture we find for both chemisorption and physisorption a final loop distribution Ω(s) ∼ s −11/5 and density profile c(z) ∼ z −4/3 whose forms are the same as for equilibrium layers. In contrast to equilibrium layers, however, the statistical properties of a given chain depend on its adsorption time; the outer layer contains many classes of chain, each characterized by a different fraction of adsorbed monomers f . Consistent with strong physisorption experiments, we find the f values follow a distribution P (f ) ∼ f −4/5 .