In this paper, we prove two normality criteria for a family of meromorphic functions. The first criterion extends a result of Fang and Zalcman [Normal families and shared values of meromorphic functions II, Comput. Methods Funct. Theory, 1 (2001), 289-299] to a bigger class of differential polynomials whereas the second one leads to some counterexamples to the converse of the Bloch's principle.
