While Kevin Costner is not my favorite actor by any means, "Robin Hood: Prince of Thieves" wasn't a bad film, at least in my memory. During the "inspirational speech" portion of the film Robin tells the disbelieving peasants "Even a child can be taught to find the chinks in armor!"
Robin was right, all armor has weak points.
With the proliferation of body armor, commercial and surplus, it makes sense to think about defeating body armor with the tools you have. So far only limited crime has been committed by criminals in body armor, but I can only see that number going up as it is part of the Zetas MO. In the criminal arms race, body armor is the next step for organized crime.
Modern armor is usually a vest with ballistics inserts and a ballistic helmet. This armor is specifically designed to protect against the "center of mass shot" that most of us train to deliver. That being said, you don't have to take the center of mass shot. Specialized armor such as an EOD suit is unlikely to be used by an attacker as it is unwieldy (however improvised armor as used by the North Hollywood Bandits covered more area than standard commercial body armor).
Generally the soft portion of the vest is multi layered kevlar, a polyarylamine with very high tensile strength. The ballistic insert can be made of ceramic, plastic, or metal (AR500 steel makes dandy inserts if you don't mind the weight). Each insert has pros and cons. Ceramic is great for one or two hits. Plastic stops hypervelocity rounds from rifles, but not always low velocity high momentum rounds from big pistols. Metal inserts are damn heavy and interfere with a compass for dismounted movement.
A ballistic helmet is generally made of kevlar fiber stiffened with a synthetic resin although there are many more surplus "steel pots" on the market than kevlar. The steel pots can be defeated by nearly any firearm, but the kevlar ballistic helmets require something in the centerfire rifle category. Spider silk has been considered a future replacement for Kevlar, but despite advances in manufacturing technology it hasn't matured as a technology.
Tools for defeating armor.
Doesn't matter how much body armor you are wearing if a tank rolls over you. That is why tankers call infantrymen "crunchies." I'm pretty sure that a soccer mom SUV will do sufficient damage to stop body armor clad criminal. The big armored "battle wagons" that the Cartels have been making south of the border will definitely crush someone in a ballistic vest.
BOMBS: Specifically IEDs
I mention this only because it is something that has become a reality on the battlefield. First in Iraq before being exported to Afghanistan this is a tool/tactic that will not go away. While I don't consider myself a jack booted thug I am plenty sure that is exactly how the goat humping jihadists see me, so their preferred method of defeating armor is a really big bang. It works.
The drug Cartels have adopted the IED as a tactic in Central America, and Brownsville, Texas, is the first place I could find where an IED was used on US soil.
Dowsing someone in a burning fluid defeats body armor. Molatov cocktails anyone? It isn't like you'll have access to white phosphorous or napalm bombs, and surplus flame throwers are getting rare as hens teeth....
While the civilized world continually disarms itself of chemical munitions a lot of tin pot dictators are getting their grubby hands on all the chemical weapons they can find. I don't recommend using chemical weapons to defeat body armor. If you you want to understand why chemical weapons are a really bad idea, http://www.ncbi.nlm.nih.gov/pubmed/10408124
GUNS: use a RIFLE.
People have eyes, and eyes need unobstructed line of site to function. So until they make bulletproof face shields (like the NH Bandits improvised), the triangle created by eyes and nose is a good place to aim if you can. If your target is looking at you and you hit anywhere in that triangle then you can move on to the next target. Disrupting the central nervous system is a good way to stop an attack, and shooting glasses aren't designed to actually be shot.
Another thing that armor wearing criminals have is joints. Specifically hip joints. Shoot someone there and if you don't kill them, then you should be able to immobilize them. The hip area represents a sizeable target, although much smaller than "center of mass." The North Hollywood Shooters covered themselves in body armor, but had the police used AR-15's with high velocity bullets against the shooters hips they could have shortened the engagement nicely. Ballistic inserts don't cover joints just yet, but I have some pretty good ideas on how to incorporate that feature into the next generation of body armor.
Branch arteries in the neck, legs, arms, are also good things to disrupt. Unless you make your own armor like the NH Bandits did (or are wearing an EOD blast suit), some limbs will be exposed. Lethality is achieved by disrupting the flow of oxygen to the brain, and cutting through an artery or two is a relatively quick way to achieve that disruption.
Another thing to consider is that a kevlar vest has holes. A torso hole at the bottom, a neck hole at the top, and arm holes on each side. Each of these holes is a good target if you have an angle to shoot it. This means elevated shooting down for the neck and arms, and depressed shooting up for the torso hole. Had the Police got off the street with a 308 caliber patrol rifle and shot down the NH Bandits neck hole from a rooftop it would have stopped the fight much more quickly.
In the extreme case of the North Hollywood Bandits, they made armor to cover their entire bodies. Good for them but in the end they still died from multiple gunshot wounds. Pistols turned out to be relatively useless, shotguns only slightly less worthless, and patrol rifles were in short supply. A 308 or 223 rifle would have punched easily through the layers of kevlar. Guess what the North Hollywood Police did not have?
Finally, there is a tactic that I have never personally used, but some hunters have called "breaking down" an animal. Bullets are shot not to kill the animal but immobilize it, aiming for the knees, hips, spine. If your bullets cannot penetrate the armor, they can at least transfer kinetic shock through it to crack bones, burst blood vessels. This is the only way to really describe what happened to the North Hollywood Bandits, the police just shot them until enough armor was damaged that bullets started getting through.
The great thing about direct fires, if you aren't achieving the effect you desire, just pour on more. It worked for the North Hollywood Police.