ZEILBERGER, A Maple package: it implements the new-improved (and simplified) Zeilberger algorithm for single-sum definite hypergeometric summation. It does what zeil does in EKHAD and Zeilberger in the built-in Maple package SumTools[Hypergeometric]. The article ”Sharp upper bounds for the orders of the recurrences output by the Zeilberger and q-Zeilberger algorithms” is accompanied by Maple packages ZEILBERGER and qZEILBERGER that implement the simplified Zeilberger and q-Zeilberger Algorithms, described in the article.