We can thank Charles Goodyear for his perseverance. In 1844, after years of experimentation, Charles Goodyear finally invented the process of heating natural rubber mixed with sulfur to create shoe rubber.
He called it vulcanization, and this is still how we make shoe rubber today. With the right combination of mixing, heat, and curing agents, sticky natural rubber is transformed into tough and colorful rubber shoe bottoms.

Vulcanization is how all the big shoe brands like Nike, Adidas, Yeezy, Vibram, Margom, and Jordan made outsoles. The molds and processes are all about the same. Black rubber, crystal rubber, or gum rubber is all a matter of the recipe. Black outsole rubber is loaded with carbon, Crystal rubber is 100% synthetic, while Gum rubber is a mixture of natural and synthetic rubber compounds.

How Rubber Soles are Made for Shoes

To make a rubber outsole you need a sole design, metal outsole tooling, a heated hydraulic press, and rubber compound.  

The rubber pressing operations for shoe soles are usually sub-contracted to a factory that specializes in the compounding and pressing of the rubber parts. The rubber mixing operations are messy and pressing rooms are uncomfortably hot, making it best done outside of the shoe factory.
Also, the rubber pressing factory will have many customers, allowing the presses to run higher volume efficiently.

See a rubber pressing factory
The rubber pressing operation begins with the mixing of the rubber components. Depending on the performance requirements of the rubber outsole, natural and synthetic rubbers are combined with curing agents such as sulfur or peroxide.