What approach did Rudolf Bultmann take when interpreting the Synoptic Gospels? What impact has his approach had on current scholarly views on the Synoptics?
Bultmann's major book on the topic (which I have not read) is The History of the Synoptic Tradition, in which he employs the form-critical method (see the title of this work by him). According to Wikipedia he is one of the pioneers of this method. John Webster says,
This book analyzes the various literary forms of the accounts of the ministry of Jesus in the synoptic gospels (miracle stories, controversy sayings, prophetic utterances, parables, etc.), and seeks to show how these forms have been radically altered, and often created, by the activity of the early church. The conclusion from the book is that the synoptic accounts of Jesus are almost useless as an historical record.
In this de-emphasis or dismissal of the question of the historical Jesus, he followed Barth, though ironically some of Bultmann's disciples revived the question (see The God Who is There in The Francis A. Schaeffer Trilogy, esp. pp. 86-87 and Donald Guthrie "Some Recent Books on the Gospels," Vox Evangelica 4 (1965): 43-54)). Guthrie says:
Although Bultmann must be regarded as the leading representative of the more extreme school of form critics in that he does not attach historical significance to much of the material, if indeed it can be said that he attaches historical significance at all, his approach has had wider repercussions than his own school of thought.
Rather than holding to a historical resurrection, Guthrie explains that Bultmann holds "that Christ’s resurrection occurs first in the believer's existence." He also says, "Bultmann's exegesis is dominated by his demythologization presuppositions." Read Webster and Guthrie yourself for further discussion. See also this article.
(This question is probably not unrelated to the Christology of Dietrich Bonhoeffer on Christianity.SE.)