FPGAs or Field Programmable Gate Arrays are wonderful microchips, that allow for arbitrary logic circuit design at the hardware level. They can be programmed to do any kind of digital processing that you can think of.
Consider that you can reuse an FGPA to be something as simple as a signal inverter, sensor trigger, or multiplexer, all the way up to recreating a modern CPU such as the Amtel chip found in Arduinos. So yes, FPGAs can do anything a microcontroller can do, and more, if you have a large enough microchip. Unlike traditional microcontrollers and CPUs, FGPAs can be massively parallel, and run without a system clock.
We are introducing the Fipsy FPGA as a lightweight way to learn about FPGAs with a low cost, and learning materials targeted at Makers.
If you have resources that can teach Verilog and introductory FPGA concepts, please e-mail them to email@example.com. We will try to create lists and examples.