A delayed coking process for making substantially free-flowing coke, preferably shot coke. A coker feedstock, such as a vacuum residuum, is heated in a heating zone to coking temperatures then conducted to a coking zone wherein volatiles are collected overhead and coke is formed. At least one polymeric additive is added to the feedstock prior to it being heated in the heating zone, prior to its being conducted to the coking zone, or both.