  In this paper we obtain new effective results on the Halpern iterations of
nonexpansive mappings using methods from mathematical logic or, more
specifically, proof-theoretic techniques. We give effective rates of asymptotic
regularity for the Halpern iterations of nonexpansive self-mappings of nonempty
convex sets in normed spaces. The paper presents another case study in the
project of {\em proof mining}, which is concerned with the extraction of
effective uniform bounds from (prima-facie) ineffective proofs.
