We consider triangular Pólya urns and show under very weak conditions a general strong limit theorem of the form Xni/a_ni^a.s.⟶Xi, where X_ni is the number of balls of colour i after n draws; the constants a_ni are explicit and of the form n^αlog^γn; the limit is a.s. positive, and may be either deterministic or random, but is in general unknown.

The result extends to urns with subtractions under weak conditions, but a counterexample shows that some conditions are needed.

For balanced urns we also prove moment convergence in the main results if the replacements have the corresponding moments.

The proofs are based on studying the corresponding continuous-time urn using martingale methods, and showing corresponding results there. In the main part of the paper, we assume for convenience that all replacements have finite second moments; in an appendix this is relaxed to L^p for some p>1.