Examples of Infinite Simple Groups

I would like a list of infinite simple groups. I am only aware of $A_\infty$.

Any example is welcome, but I'm particularly interested in examples of infinite fields and values of $n$ such that $PSL_n(F)$ is simple.

References about this topic, or any example, are also appreciated.

A Tarski monster group is a finitely generated, infinite group where every proper, non-trivial subgroup is cyclic of order a fixed prime $p$. These were shown to exist for all $p>>1$ in the 80s by Ol'shanskii. Moreover, they are simple groups.

To see that Tarski monster groups are simple, suppose $N$ is a normal subgroup of a Tarski monster group $G$. Then pick some proper subgroup $M\neq N$. As $N$ is normal, $MN$ is a subgroup of order $p^2$, a contradiction.

Richard Thompson's groups $T$ and $V$ are well-known examples of infinite simple groups. See this answer of mine for more details, or look up the article Introductory notes on Richard Thompson's groups by Cannon, Floyd and Parry. They are defined by their action on the unit circle.