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,
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:
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.)