Let $\mathbb{F}_q$ be the finite field with $q$ elements. $\mathcal{X}$ denotes an absolutely irreducible, projective curve defined over $\mathbb{F}_q$. Having many applications in other branches of mathematics, special interest arises on the question how many rational points (i.e., the points with coordinates in $\mathbb{F}_q$) $\mathcal{X}$ can have. Hasse and Weil showed that the number $N(\mathcal{X})$ of rational points of the curve $\mathcal{X}$ is bounded by $q$ and an invariant $g(\mathcal{X})$ attached to the curve (which is called \textit{genus}); namely $$N(\mathcal{X})\leq 1+q+2g(\mathcal{X})\sqrt{q} \ .$$ This bound is called the \textit{Hasse--Weil Bound}. Then Ihara and Manin observed that this bound is not optimal when the genus is large compared to the cardinality of the finite field $q$. This observation led to the investigation of the number of rational points of curves of large genus, and resulted in \textit{Ihara's constant} $A(q)$ defined by \begin{align*} A(q):= \mathrm{limsup}_{g(\mathcal{X})\rightarrow\infty}\frac{ N(\mathcal{X})}{g(\mathcal{X})} \ , \end{align*} where $\mathrm{limsup}$ is taken over all curves defined over $\mathbb{F}_q$ with genus tending to infinity. In this talk, I will briefly describe curves over finite fields, their number of rational points, Ihara's constant and discuss recent developments.