For a country like Iran under sanctions of US and most industrial countries, the first question you are going to ask is "Is it safe to travel Iran now?" It is the question that most of millions of travelers to Iran have already answered and the answer is: it is safe as anywhere else in the world.
Many airlines have direct flights to Tehran. It includes British Airways, Lufthansa, Austrian Airlines, , Aeroflot, Alitalia, Swiss, Turkish Airlines, Air India, and a large number of middle east carriers like Iran air, Mahan air, Emirates, Etihad Airways, Air Arabia, Gulf Air provide fast and easy access to Iran with their really low cost airfares to and from Tehran.
Direct flights to Iran are possible from London, Paris, Frankfurt, Milan, Madrid, Hamburg, Geneva, Vienna, Tokyo, Seoul, Beijing, Mumbai, Kuala Lumpor, Dubai, Doha and lots of other major cities in Europe, Asia, and Middle East.
IranAir is the national airline of Iran and offers flights to Iran from 17 European airports. German flights to Iran operate from Cologne, Frankfurt and Hamburg whilst English and French flights to Iran operate only from the two capital cities, London and Paris respectively. Flights to Iran also operate from Moscow, Geneva, Amsterdam, Istanbul, Ankara, Vienna, Stockholm, Moscow, Copenhagen and Larnaca.
Mahan Air is the largest private airline in Tehran. Mahan Air has flights to and from Tehran to Baghdad, Istanbul, Dubai, India, Dusseldorf, Birmingham and a couple of other european cities.
Most flights to Iran land in Iran Khomeini Airport (IKA). IKA is a large and modern Airport that has been opened in 2004 and all international flights have been transferred to it from the older Mehrabad International Airport.
It's considered wise to look compare the price slots of airlines and then accordingly get lowest airfares. Best way is to compare, compare and compare before picking your best airfare deal.