Technologies called "suspensions" support the comfort of passengers in wheeled vehicles. In the era of draft animals, carriage suspensions apparently started with hanging the passenger compartment by iron chains; then replacing the chains with less noisy leather straps; then mounting the same on an iron elliptical leaf spring; then stacking several leaf springs; and today's standard for passenger vehicles is coil springs.
When were these advances developed and popularized?
The first suspension system has been designed for the light chariots of Ramses around the year of 1296 B.C.
hanging the passenger compartment by iron chains or leather straps
It is likely that Roman carriages (1 BC) employed some form of suspension on chains or leather straps, as indicated by carriage parts found in excavations.
Leaf springs have been around since the early Egyptians. Ancient military engineers used leaf springs in the form of bows to power their siege engines, with little success at first. The use of leaf springs in catapults was later refined and made to work years later. Springs were not only made of metal, a sturdy tree branch could be used as a spring, such as with a bow. Horse-drawn carriages and the Ford Model T used this system, and it is still used today in larger vehicles, mainly mounted in the rear suspension
The venerable leaf spring, which some manufacturers still use in rear suspensions today, was invented by Obadiah Elliot of London in 1804. He simply piled one steel plate on top of another, pinned them together and shackled each end to a carriage.
Coil springs first appeared on a production vehicle in 1906 by Alanson Brush in the Brush Runabout made by the Brush Motor Company. Today, coil springs are used in most cars.