By the late 15th century, the earliest viable handheld firearms in Europe were the arquebus fitted with a matchlock mechanism.
With the advent of the heavy arquebus (later known as a musket) in the early 16th century, a spout large enough to measure the required powder was impractical.
A bandolier from whence would hang several little flasks each filled with a dedicated premeasured charge was the solution to this problem.
A somewhat different form of bandolier came into use in the late 19th century that held modern metallic cartridges and hand grenades.
In fact, some aftermarket shotgun slings are designed in a fashion similar to traditional bandoliers, albeit with a far more limited capacity.
The Chetniks of World War II made heavy use of bandoliers, often carrying two over the shoulder and two around the hip, using the latter as a holster for revolvers and daggers.