ARM processors handle exceptions and interrupts by mapping them to different processor modes and using an exception vector table to point to the corresponding handler addresses. The highest priority exceptions are reset, data abort, and FIQ. IRQ is the second highest interrupt. When an exception or interrupt occurs, the processor saves context by storing CPSR to SPSR and PC to LR, then loads the handler address and jumps to the exception mode. Handlers preserve context, service the request, then return using LR or stack to restore context and resume original execution mode.