In this paper we establish a number of new lower bounds on the size of a critical set in a latin square. In order to do this we first give two results which give critical sets for isotopic latin squares and conjugate latin squares. We then use these results to increase the known lower bound for specific classes of critical sets. Finally, we take a detailed look at a number of latin squares of small order. In some cases, we achieve an exact lower bound for the size of the minimal critical set.