Since some theories are powerful enough to model different mathematical objects, it is natural to wonder about their own consistency.
Gödel's incompleteness theorems show that Hilbert's program cannot be realized: if a consistent computably enumerable theory is strong enough to formalize its own metamathematics (whether something is a proof or not), i.e. strong enough to model a weak fragment of arithmetic (Robinson arithmetic suffices), then the theory cannot prove its own consistency.
For theories at the level of second-order arithmetic, the reverse mathematics program has much to say.
The method of forcing allows one to show that the theories ZFC, ZFC+CH and ZFC+¬CH are all equiconsistent (where CH denotes the continuum hypothesis).
The consistency strength of numerous combinatorial statements can be calibrated by large cardinals.