Conditions are established under which maximum likelihood estimators are consistent and asymptotically normal in the case where the observations are independent but not identically distributed. The key concept employed is uniform integrability; and the required convergence theorems which involve uniform integrability, and are of independent interest, appear in the appendix. A motivational example involving estimation under variable censoring is presented. This example invokes the full generality of the theorems with regard to lack of i.i.d. and lack of densities $\operatorname{wrt}$ Lebesgue or counting measure.