United States, United Airlines offers nonstop daily service from Newark Airport to Delhi and Mumbai; Air India offers daily non-stop service to Delhi from JFK and ORD and Mumbai from EWR. American Airlines offers nonstop daily service from Chicago to Delhi. Various European airlines offer connecting service through their European hubs from most major US cities and various Asian airlines offer connecting service from West Coast cities to India through their Asian hubs. Jet Airways also flies from New York to Delhi, Mumbai, and Chennai via Brussels.
Entries from Europe and Northern America are possible using many European airlines such as Lufthansa, Finnair, British Airways, KLM Royal Dutch Airlines, Air France and Virgin Atlantic. For long-term visitors (3-12 months), Swiss airlines often have good deals from Switzerland with connecting flights from major European and some American cities as well.
To save on tickets, consider connecting via Gulf countries, by Air Arabia (Sharjah-based low cost carrier having some connections in Europe), Etihad (especially if you need one-way ticket or going back to Europe from another Asian country) via Abu Dhabi, as well as Emirates via Dubai or Qatar airways via Doha. Obviously, these airlines are also the easiest way to come from the Gulf countries themselves, along with Air India and Air India Express.
From East Asia and Australia, Singapore (which is served by Air India, it's low-cost subsidiary Air India Express, Jet Airways, as well as Singapore Airlines, it's subsidiary Silk Air and low-cost subsidiary Tiger Airways has arguably the best connections to India with flights to all the major cities and many smaller ones. As about the cheap way from South-East Asia or vice versa, Malaysian low-cost carrier AirAsia is usually the best choice (if booked well in advance, one-way ticket price is normally below US$100, sometimes being less than US$50, they have connections from China,