Darby International is a major international law firm, owned and managed by Edward Darby. Not much is known about this firm. It was first implicitly mentioned in the episode "Play the Man" when Dana Scott came for merger negotiations with Harvey Specter. It is known though that the firm is very rich and has vast amounts of resources, as Harvey Specter wanted them to fund Pearson Hardman's class action lawsuit after Daniel Hardman made sure they had no funding left.
Looking well towards the end of Season 3, the firm discovers Stephen Huntley was responsible for the killings, which brought charges against Ava Hessington. After Harvey and Edward were successful in convincing Cameron Dennis to settle, Jessica, Mike and Harvey, used the opportunity to secretly place an agreement forbidding Darby to practice law in the United States. Having no choice, Darby signs the agreement and the dissolution of Pearson Darby Specter began, after which the London office reverted back to Darby International.