Minimal Markov bases of configurations of integer vectors correspond to minimal binomial generating sets of the assocciated lattice ideal. We give necessary and sufficient conditions for the elements of a minimal Markov basis to be (a) inside the universal Gr{ o}bner basis and (b) inside the Graver basis. We study properties of Markov bases of generalized Lawrence liftings for arbitrary matrices $Ainmathcal{M}_{mtimes n}(Bbb{Z})$ and $Binmathcal{M}_{ptimes n}(Bbb{Z})$ and show that in cases of interest the {em complexity} of any two Markov bases is the same.