Though GE often gets the credit of starting it all here, it's actually American Express which pioneered offshore processing out of India. Way back in 1994, its decision to start what was then called the Financial Center East (FCE) in Gurgaon, as part of consolidating its 46 processing centers in three global locations, was clearly ahead of time.
Unlike most others, AmEx India, however, did not start as a back office for the American operation. Rather, the India center was established in 1994 to serve Japan and Asia Pacific. However, by 2003, only 58% of the work was for Japan/AsiaPac, 20% for Europe, 24% for US, and 8% for Latin America. From 1995, when it first migrated a process, to 1999 when it added high-end capabilities like financial analysis and modeling, it has constantly moved up the value chain. The company has not just saved cost by labor arbitrage but also by saving time for invoicing payout to vendors, which has resulted in larger discounts. Last year, it began a pilot for serving UK, Australia, and Canada as new geographies, according to an E-Business Strategies report. 
American Express (India) also handles vendor relationships in India. It is one of the most prolific outsourcers to third-party vendors in India.