  We show that the complexity of the Markov bases of multidimensional tables
stabilizes eventually if a single table dimension is allowed to vary. In
particular, if this table dimension is beyond a computable bound, the Markov
bases consist of elements of Markov bases of smaller tables. We give an
explicit formula for this bound in terms of Graver bases. We also compute these
Markov and Graver complexities for all $K \times 2 \times 2 \times 2 $ tables.
