The term Far East usually refers to East Asia, including Mongolia, China, Korea, Japan, and Southeast Asia. It sometimes includes the Philippines, Indonesia, and the other Oceanic countries, and it sometimes includes eastern Russia and the Indian subcontinent. The Near East is the eastern Mediterranean region once dominated by the Ottoman Empire. Middle East—the newest of the three terms—originally referred to everything between the other two Easts (Mesopotamia to Burma), but it now usually denotes the Near East in addition to Afghanistan, Iran, and the Arabian peninsula.
All three terms are vague, and their Eurocentrism makes them easy to criticize. Dropping them from the language would be a no-brainer if they weren’t so widely used. Plus, many of the alternatives are even worse (except East Asia, which has gained ground in place of Far East). So Middle East (sometimes shortened to Mideast) and Far East are still deeply entrenched in the language and are used across the English-speaking world. Near East is obsolescent, however, as our modern definition of Middle East covers much of what used to be considered the Near East.
For example, these major news publications have seemingly no qualms about using the vague and Eurocentric Middle East and Far East:
Recent events across the Middle East show that the desire for justice and human dignity is universal. [New York Times]
Thomas had spent 30 years scouring Latin America, the Far East and most other points around the globe. [Guardian]
Wars in the Middle East, tax cuts for the wealthy and unfunded social welfare programs have the US on the brink of defaulting on its $14.3 trillion debt. [Sydney Morning Herald]
Toy makers and collectors in the Far East as well as stateside are among Robertson’s fans. [Boston Globe]