Nowadays this question can seem odd. Computer system software architectures are heavily depending on interrupts and systems are far too complicated to build up without interrupts.

But if we go back in time some 40 years we had simple computer systems without interrupts. Some programmable automation systems. One program (we can call it operating system or application) was looping all the time in one big loop and doing some tasks whole time. For example:

Code:
loop( forever)
{
read (keyboard buffer); if(keyboard pressed) write (char to display); read (pressure measurement1) ; if(pressure measurement1 too high) open_valve(1); read(temperature1) if(temperture1 too low) ................
}
So, we were replacing interrupts with polling.