Let us suppose that $G$ is a Lie group.
The space of maps $map(S^1,G)$ is a topological group called a loop group, maybe it is better to consider smooth loops, in that case we have an infinite dimensional Lie group. A very nice and classical reference is
Pressley, Segal "Loop groups", Oxford Mathematical Monographs.

From the point of view of string topology, I can say a few things:

R. Hepworth, L. Menichi, and S. Kupers have done some computations (I am very sorry if I have forgotten someone). The computation of the Chas-Sullivan BV-structure of the loop homology is very fun.

In the case of the Lie group $U(n)$, you can consider spaces of polynomial loops and they have a nice geometric filtration given by the polynomial degree. This filtration is related to Morse theory of loop spaces associated to the energy functional. The Chas-Sullivan BV-structure is compatible with this polynomial filtration.

Computations of these homology groups can be very useful if you want to compute the BV structure of the loop homology of $\mathbb CP^n$ and $S^n$. Chas-Sullivan operations give some informations on closed geodesics you can have a look at Goresky-Hingston's paper
"Loop products and closed geodesics"
Mark Goresky and Nancy Hingston Duke Math. J. Volume 150, Number 1 (2009), 117-209.