A software defined radio can theoretically be made out of little more than a microprocessor with an onboard ADC and some DSP code. This is exactly what Alberto di Bene (I2PHD) achieved by connecting an antenna directly to the on board 12-bit ADC on a STM32F429 Discovery board.
To make it actually work as an SDR he also wrote some code to utilize the development board’s ARM processor which processes the ADC input into a radio signal, demodulates it and then turns it into audio via the boards DAC and speaker. The radio can tune from 8 kHz up to about 900 kHz.
The only real extra hardware in Alberto’s system is a low pass filter for anti-aliasing and impedance transformation, and a reconstruction filter to get sound to the speakers from the DAC. He also used the boards LCD screen to implement a full GUI tuning system.
A PDF document detailing his work can be downloaded here.