If you qualify as an American citizen residing abroad (basically having lived at least one year abroad), there are two methods by which you can reduce your US tax by a substantial amount. These are the "Foreign Earned Income Exclusion (FEIE)" and the "Foreign Tax Credit."
However, neither of these methods excuses you from filing if your income was above the filing threshold.
The Foreign Earned Income Exclusion (FEIE, using IRS Form 2555) allows you to exclude a certain amount of your FOREIGN EARNED income from US tax. For tax year 2018 (filing in 2019) the exclusion amount is $103,900. What this means is that if, for example, you earned $118,000 in 2018, you can subtract $103,900 from that leaving $14,100 as taxable by the US. But beware: this $14,100 is taxable at tax rates applying to $118,000 (the so-called "stacking rule"). The exclusion applies only to foreign earned income. Other income, such as pensions, interest, dividends, capital gains, US-sourced income, etc., cannot be excluded with the FEIE. You are liable for full US tax on this type of income.
Here's a simple example. Suppose you live in France and you earned EURO 100,000 (about $118,000) from your French employer. You are married filing jointly, have two children and you take the standard deduction ($24,000) and child tax credit ($4,000 for two children).
The US tax on this income is calculated as follows:
US tax on $118,000 is $8,559
US tax on $103,900 (amount excluded) would be $5,457
Net US tax payable
($8,559 - $5,457) = $3,102
While this is only an approximate calculation, it gives you an idea of how the system works.
The other method for reducing your US tax bill is the foreign tax credit, using IRS Form 1116. If your income was taxed by a foreign country, you can subtract that tax from your US tax, in most cases substantially reducing your US tax bill. But be careful: you cannot claim a foreign tax credit for foreign taxes on income excluded on Form 2555. In other words, you can only claim a foreign tax credit for foreign taxes on the same income that the US is taxing. The fraction of your foreign taxes that can be taken as a tax credit is determined by the ratio of non-excluded income to total income. Here's an example, using the same figures as above.