Summary: This paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains [see the author, Am. Math. Mon. 82, 813–817 (1975; Zbl 0315.30036
)]. These include short and efficient proofs of generalizations of (i) the Picard Theorems, (ii) Gol’dberg’s Theorem (a meromorphic function on
which is the solution of a first-order algebraic differential equation has finite order), and (iii) the Fatou-Julia Theorem (the Julia set of a rational function of degree
is the closure of the repelling periodic points). We also discuss Bloch’s Principle and provide simple solutions to some problems of Hayman connected with this principle.