[Capacitive power supply circuits][1] have been used for decades in things like alarm clocks, small appliances (coffee makers) and the like. They work pretty well for small constant loads without the bulk of a heavy transformer.

[![enter image description here][2]][2]

**However, they are scary to work on and need special considerations for user safety.**

**Do not think about using one unless you really know and understand what you are doing.** 

Even an experienced engineer like myself will shy away from them, designing everything to the right of the bridge using a bench supply till everything else is proven functional before plugging in the AC with my other hand in my back pocket.

**ADDITION**

Some switching power supplies also avoid using the large power transformer by rectifying the mains voltage into a large DC voltage then use high frequency switching through a much smaller transformer. Again, though, "There be dragons here!", and the DC voltages on the primary capacitors are nasty.

  [1]: http://www.electroschematics.com/5678/capacitor-power-supply/
  [2]: https://i.sstatic.net/czCeb.png