From the UK:
There are flights from most UK airports to Istanbul Ataturk International Airport, not far from the city. The flight duration from London to Istanbul is approximately 3 or 4 hours.
British Airways flies to Istanbul from Heathrow, Aberdeen, Edinburgh and Glasgow, while Turkish Airlines has cheap flights leaving from Heathrow, Stansted, Manchester, Birmingham and Dublin. Swiss also offers low-cost flights to Istanbul from Heathrow and Manchester.
From the USA:
There are direct flights from the USA to Istanbul Ataturk International Airport, which is situated about 15 miles (23km) west of the city centre.
There are direct cheap flights to Istanbul from New York and Chicago available on Turkish Airlines. KLM and Delta also fly to Istanbul direct from New York, and flights on United Airlines depart from Chicago.