Let $A$ be an infinite non-empty subset of $\mathbb{N}$. For each $n\in\mathbb{N}$, define $r_{A,A}(n):=|\{(a,b):\,a,b\in A,\ a+b=n\}|$ and $R_{A,A}(n):=\sum_{j\leq n}r_{A,A}(j)$. These are two kinds of basic additive representation functions. The study of additive representation functions have a long and storied history. One of the most famous conjecture of representation functions is the Erdös-Turán conjecture for additive bases, namely, if $ r_{A,A}(n)\geq 1$ for all sufficiently large $n$, then $ \limsup_{n\to\infty}r_{A,A}(n)=\infty$. In this talk, some previous and classical results concerning the additive representation functions will be retrospected. In addition, our recent progress on this topic will be introduced.