Modern soldiers still wear plate armor. Thing is, it's quite heavy depending on the material used (mostly steel, or ceramic), because they have to be rated to absorb the shock of certain calibers, typically multiple times before failing. So, for pistol calibers up to 9mm, you can mostly get away with just wearing a thin kevlar sheet, but rifle calibers require either a steel or ceramic plate.
The challenge to creating modern armor, however, will always be a simple equation: momentum. Simply put, you need to have enough mass to decelerate a bullet, otherwise that energy is going to travel through the armor and still injure the person wearing it. You can see this pretty clearly when it comes to kevlar body armor; although it's incredibly light, that lightness is a bit of a double-edged sword, because it means the energy of the bullet must be mostly absorbed by the wearer. Depending where the shots land, this could mean severe bruising to broken bones. Admittedly, broken bones and bruising is a hell of a lot better than being dead, so most people are fine with this trade-off. In other words, because we relying on physically decelerating projectiles, armor literally has to be heavy, otherwise it wouldn't work.