My favorite PIC Microcontroller projects are clock projects. All of my clocks are powered by 5V DC wall wart adaptors. In case of power failure, the clocks stop and I have to set time when the electric power is restored. This problem will not happen with the clocks using DS1307 RTC as it provides battery backup capability. However, the clocks without DS1307 or any other RTCs will suffer from power failure. The circuit below is an example of the use of battery backup circuit with PIC16F627A or PIC16F628. The circuit can be used with other Microcontroller chips as a battery backup source. It also provide power failure signal (active low) for further processing.
