  We prove various congruences for Catalan and Motzkin numbers as well as
related sequences. The common thread is that all these sequences can be
expressed in terms of binomial coefficients. Our techniques are combinatorial
and algebraic: group actions, induction, and Lucas' congruence for binomial
coefficients come into play. A number of our results settle conjectures of
Benoit Cloitre and Reinhard Zumkeller. The Thue-Morse sequence appears in
several contexts.
