In this work, we undertook an exploratory study of QCD thermodynamics with domain-wall fermions. This had been studied before but with much smaller lattices and a heavier pion. In this new study, we report on results obtained on much larger volumes and discuss what needs to be done to go even closer to the chiral limit. A second new aspect of our study was the introduction of a chemical potential for the first time in the domain-wall formalism. We measured the lowest-order quark number susceptibilities and found a well-defined, smooth transition in some of these susceptibilities.We also carried out several analytic calculations in the free-field case. One motivation was to understand how these fermions worked, especially with respect to thermodynamic simulations. However another motivation was to understand cutoff effects, which had not been studied before for these fermions.We found that these effects were significant for the free operator. We therefore implemented and tested an improved version of the operator with much smaller discretization errors. In this work, we also present some preliminary results of simulations that show that observables measured using this operator show much smaller cutoff errors.