(This question also applies to other Game Boy games, but Pokémon seems to be one of the best-documented.)
Pokémon Red & Blue (Red & Green in Japan) is a pair of Game Boy games about capturing "Pocket Monsters", or "Pokémon", by fighting them with other Pokémon. Since its release lots of bugs and glitches have been found, many of which are caused by overflow errors.
The Game Boy contained a Sharp LR35902 CPU, a Z80 derivative that was something of an intermediate design between the Z80 and 8080:
This processor is similar to an Intel 8080 in that none of the registers introduced in the Z80 are present. However, some of the Z80's instruction set enhancements over the stock 8080, particularly bit manipulation, are present. Still other instructions are unique to this particular flavor of Z80 CPU.
Both the Intel 8080 and, by extension, the 8080-compatible Zilog Z80 have a carry bit that is set on operations that overflow. A derivative of these chips would also have these; indeed this is not documented as a difference between the Z80 and the LR35902.
To the part that is confusing me: the LR35902 CPU had an overflow flag, and there are many exploitable glitches in Pokémon Red that are caused by overflows; there are glitches caused by overflows in most places that overflows can occur. There must be a reason that the developers didn't use this flag on so many occasions, but I can't work out what it is.
Why are there so many overflow bugs in Pokémon Red?