I know espionage from the Eastern bloc to the West was mainly KGB. For example, the Soviets stole secrets from the U.S. during the space race. Did the Stasi have significant influence in this area as well? Or did the Stasi focus mainly on domestic espionage like surveilling the GDR's people?
The Stasi had significant success in penetrating the West, not necessarily to the same degree as the Soviets, but given their comparative lack of resources, they were very effective.
Probably the most famous Stasi penetration of the West was their insertion of Gunther Guilliame, a Stasi agent, as personal advisor to Willy Brandt, West Germany's left wing Chancellor. Brandt's controversial policy of Ostpolitik, or the opening up of diplomatic and commercial ties with Eastern Europe, was popular both in Germany and in the USSR, but Brandt was forced to resign when Guilliame fled to East Germany. Ironically, Stasi Director Markus Wolf would later say that Guilliame's exposure was an "own goal" for the Stasi, since they wanted to keep Brandt in power.
Possibly more damaging was Rainer Rupp, aka Topaz, who worked for NATO in SHAPE HQ in Brussells and passed critical strategic information to the Stasi, including the deployment locations of Pershing missiles in Europe.
In a recent article Der Spiegel reported that there were nearly 2000 West Germans reporting to the Stasi in December 1988. Many of them were quite low level and provided non-strategic intelligence.
'A Man without a Face', Markus Wolf