Set up daily or weekly price alerts.

Don't buy more than 3 months in advance.  Start checking prices at that time.

Prices stay the same until about 28 days before departure.

Prices are lowest 18-28 days before departure (=magic window).

Domestic fares are lowest 21 days before departure (about 8% lower than average fares available six months before departure) (seems like a strange comparison)

Fares increase 5% (on average) two weeks before departure, 30%  the week before departure.

"International fares don’t fluctuate as much; 34 days before departure, fares are about 4% lower than the average at six months before departure."  Still, somehow I got a flight to Europe on a Turkish airlines flight that for a few hours (or maybe days) was about half the price of other airlines -- and this was about 3-4 months before departure.  So "fluctuate" might not be the right verb for a sentence talking about average.

Consider booking hotel when booking airfare.  Apparently those package deals really are good deals, and can include meals.

Airlines announce sales on Tuesdays -- look mid-day Tues to Wed. morning for best sales.

They also announce last-minute weekend deals on Tuesdays.

Wednesday is the cheapest departure date, then Tuesday, then Saturday.

For weekend trip, cheapest is to leave Saturday and return Monday.

Early morning flights are often cheaper and more reliable (fewer delays/cancellations); overnight flights, late afternoon, and dinner flights can also be good deals.

Smaller airports can be cheaper.

One stop flights can be cheaper than non-stops.

One way tickets (from different airlines, each way) can be a bargain.  See Hacker Fares.

If prices go down after you book, you might be able to get a credit.  If within 24 hours of purchase, you'll get it all back.  If later than that you might get a credit, but you might be charged a fee.