The other main way would be using a constant current driver[1]. By adjusting the amount of current to the LED, you can control the brightness.

[1] https://electronics.stackexchange.com/questions/256336/does-...

running a smaller, continuous, current through them.

PWM is done because it's cheap to implement given a fixed power rail and a digital control : need a single/couple of power switches that can even be integrated on die for low-ish power targets. continuous things usually need some kind of discrete filters, that means more components etc

Eizo has screens that use DC dimming:

http://www.eizoglobal.com/library/basics/eyestrain/index.htm...

I never tried them though.

LEDs are current devices; any mechanism for controlling current precisely can be used to control LED brightness. there are a lot of them.

PWM is just one (very inexpensive) way to implement a DAC (digital to analog converter). Other methods actually set the output voltage to the target voltage, such as a R-2R ladder. I don't think R-2R ladders are used very much in practice anymore but I can't remember any other implementations off the top of my head.