In this article, a brief history and developments on additive uniqueness sets for arithmetic functions are introduced. Let $S$ be a set of arithmetic functions. A set $E \subset \mathbb{N}$ is called an additive uniqueness set for $S$ if $f \in S$ and the condition $$f(a + b) = f(a) + f(b) \quad \text{for all } a, b \in E$$ determines $f$ uniquely. Various sets have been shown to be additive uniqueness sets and many other variations of the problem were investigated.