The chips themselves have SPI interfaces, and support programming over SPI.
Alas, not all of them.
There is also "HV Parallel programming", which is the only way to program some of the older AVRs.
And there is "HV Serial programming", which was needed on the ATtiny11 (which gained some popularity when Atmel was selling them for $0.25 each, and lost it when the chip was discontinued.)
And there are PDI amd JTAG, used on some of the newer chips.
And there is TPI, used to program the extra-tiny ATtiny4 and similar chips.
SPI progammers like the USBtinyISP will program MOST chips, though.