Zero estimates have a long classical history in diophantine approximation and transcendence,
and more recently they have been useful in counting rational points on analytic sets.
They can be considered as generalizations of the fact that a polynomial has no more zeroes than its degree.
We give an old example for exponential polynomials that is best possible,
a new example for hypergeometric-type functions that is far from best possible,
and we briefly mention some applications. We also give a counterexample for elliptic functions,
obtained with Dale Brownawell, that casts doubt on the naive meaning of best possible