The shotgun is perhaps the most notorious weapon in Project Zomboid because of the noise it creates. However, it is extremely powerful and it can potentially kill hundreds of zombies. However, its finite ammo supply can be a nuisance, and the ammo itself can weigh quite a bit.
A shotgun is probably the most powerful weapon in Project Zomboid, but also the most controversial. Although each shell can potentially kill 3-8 zombies, the noise it creates attracts more zombies. However, it is a good idea to bring a shotgun with you, but do not bring more than 20 shells with you, or you will be encumbered. The shotgun should be used only in absolute emergencies, situations where melee weapons won't cut it. If forced to use a shotgun, take two shots, sprint some distance back and shoot again. You never want to be cornered when shooting zombies, a good fallback can buy you some time to kill more zombies.
A shotgun is fairly rare, and is found in some houses. Many NPCs carry shotguns and may even have a couple dozen shells on them. Also, the raider in the tutorial "Till Death do us apart" has a shotgun, and killing him will give the shotgun and its ammo.
Putting a shotgun and a saw in the crafting area will craft a sawn-off shotgun, which has advantages and disadvantages.