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 tau ads are typically micro- to milli-seconds, for chemisorbing chains of N units we find experimentally accessible times tau 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 omega(s) approximately s(-7/5) of loop lengths s up to a maximum size smax approximately (Qt)(5/3) after time t. By times of order tau 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 Omega(s) approximately s(-11/5) and density profile c(z) approximately 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) approximately f(-4/5).