The ancient Babylonians used sharpened weapons such as sickle swords, socketed axes, spears, and the Egyptian-derived bladed mace, as well as clubs and staffs, and projectiles flung from war slings. The Babylonians were a Bronze Age people, so the bladed weapons they used were made of a softer metal than later civilization's iron and steel. This made them less effective in battle than iron and iron-alloy based weapons, as iron is harder and harder blades can take a sharper point.
At the time, though, bronze was top-of-the-line technology, and the Babylonians were so good at empire building that later armies modeled their weapons after Babylonian designs.
They did not possess bows; neither the long bow nor the cross bow have been found at excavations. Enemies such as the Akkadians did possess bows, so this weapon was not unknown to the Babylonian state. Babylonian warfare relied upon armored foot soldiers, instead of massive projectile fire or charging horsemen.
Babylonian armor was constructed out of heavy leather studded with copper and bronze. They wore breastplates made of bronze because the top of the torso is a large target and contains vital organs that need extra protection. Helmets were made out of copper to reduce weight on the head and increase comfort during battle. Fighting on foot requires a combination of heavy protection and mobility, and this was an area where the Babylonians decided to trade off defense for agility.