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The Bergman space $A^p$ consists of all holomorphic functions defined in the unit disc the $p$-th power of whose complex modulus is integrable with respect to the area measure. We say $\{z_k\}$ is an $A^p$ zero-set if and only if there exists a function $f$ in $A^p$ such that f vanishes precisely on this set. In this talk we will discuss properties of $A^p$ zero-sets. For this purpose the following three questions will be answered: Do $A^p$ zero sets vary with $p$? In other words, does there exist an $A^q$ zero-set that is not an $A^p$ zero-set for two positive real numbers $p$ and $q$? Is the union of two $A^p$ zero-sets an $A^p$ zero-set? Must every subset of an $A^p$ zero-set be an $A^p$ zero set? I will mainly follow Charles Horowitz's 1974 work in which all three questions are answered.