Q: Chemical composition of stab-proof vest?
A: Provided by JoB
There is no such thing as a stab- or bullet-proof vest. The material of choice is still Kevlar (https://en.wikipedia.org/wiki/Kevlar#Armor), because of its main characteristic, "its high tensile strength-to-weight ratio", as Wikipedia puts it. The projectile starts to push the Kevlar into the soft target a.k.a. human body, the Kevlar fibers' tensile strength prevents them from ripping, so Kevlar outside the projectile's path is pulled to press into the target. Net result: The projectile's kinetic energy gets distributed over a much larger area of the body's surface, which (hopefully) prevents the whole mess from puncturing the body.
Q: Does advanced armor exist now (or of the 20 minutes into the future variety) that could stop swords and arrows while being lighter than plate armor?
A: Provided by JoB
The ways for a firearm, or, more precisely, its projectile, to overcome a Kevlar vest are a) smaller caliber / tip and/or higher kinetic energy, and b) - to an extent - coatings that facilitate the projectile's slipping through the mesh of Kevlar fibers. (Don't put too much money on the latter. Oh, and I guess c) explosive charges are officially a viable method, too.) At some point, deflecting such a "brute force" bullet will need so many layers of Kevlar as to make the "vest" fail as clothing, or it just fails to distribute the projectile's energy over enough of an area to make the impact sufficiently harmless.
Swords and arrows are much slower but weigh more than most bullets, so, for them, a Kevlar vest's failure mode is typically the equivalent of b), the projectile slipping through between the Kevlar fibers.
As you can see, in both cases, it's not the primary property of Kevlar, its tensile strength, coming to a limit, but some secondary property. Which makes it debatable whether it's the very material that's the cause of failure, or whether discovering some stuff with (say) 10% more tensile strength per volume/weight would promise to yield a 10% improvement of protection.