I had the bad fortune of losing my passport in Belgium this trip so I thought I'd share what I noticed about the process. First, i reported the passport lost at the train station. I lost it on the train. In Belgium, the station was really too busy to help besides filling out a form. I then called the US Embassy, It was a Saturday night, but the Embassy has a emergency operator on duty to take my info. She took my info and told me to go to the Consulate on Monday during business hours and she'd send them a message that I was coming. I think it's a good idea to call ahead even if you lost your passport during business hours. These days the US Embassy and consulates are guarded blocks away and they seem to check that I phoned ahead. If you've had your Passport stolen, go to the local police and fill out a report beforehand. For a lost Passport, the Consulate said that was up to the person If they wanted to report it to the local police, but they didn't require it. Also, if you have time to get them, bring passport photos. The Consulate has a list on places that follow their expectations. The rest of the process is pretty easy. Pay a $135 fee and fill out a form. You will get a one year limited passport . I got mine in twenty minutes after being seen .They also give you a form to fill out once you get home so you can exchange the limited passport for a regular passport for free. The only reason they time limit it is because the regular passports have security features the Embassy can't reproduce.
Hope this eases some minds on how quick the process is. It's stressful enough losing your passport.