I am searching for a comprehensive survey article (or more different articles) on the subject of isoperimetric problems from ancient Greece to modern mathematical physics. Could you point out some highlights?



Survey paper on isoperimetry
There's been several articles in the comments that are "historical survey" articles. Its not totally clear if you're interested in "current research surveys," but if you are, here are several very nice ones:


Osserman's article: <http://www.ams.org/mathscinet-getitem?mr=500557> provides an excellent survey of older results. 


Ros's survey <http://www.ugr.es/~aros/isoper.pdf> contains a more modern perspective, including several very interesting open problems (for example, for dimension $n\leq8$, the isoperimetric problem in a slab is always solved by half-spheres and cylinders, while for $n\geq 10$, there are other shapes, the "unduloid" which are better for some volumes. It is unknown if this occurs for $n=9$!)


Druet has some nice notes <http://math.arizona.edu/~dido/presentations/Druet-Carthage.pdf> on the isoperimetric problem for Cartan--Hadamard manifolds (a famous conjecture suggests that in a simply connected manifold of non-positive sectional curvature, the boundary area of an isoperimetric region enclosing a fixed volume should be greater than the corresponding question in Euclidean space---this is known in dimension 2,3,4, but no higher!)


Eichmair and Metzger have collected an (amazingly short) list of all of the manifolds in which we know what the isoperimetric regions look like in Appendix H of their paper <http://www.ams.org/mathscinet-getitem?mr=3127063>


