Looking for books discussing aspects of the Geonic Period, whether it be analysis of the Responsa, a historical outline/discussion.
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From a historic perspective I am a big fan of History of the Jewish People: From Yavneh to Pumbedisa
The blurb on the Artscroll website reads:
For the first time, Jewish history is presented according to authentic Jewish sources; well researched and clearly illustrated with photos, charts, and maps. Vol. II: From Yavneh to Pumbedisa: In this volume, we trace the history of the Jews from the destruction of the Second Temple through the era of the Geonim.
You can view a good part of the book on google books here - Part 2 of the book deals expressly with the Geonic period and goes into great detail as to the historic background and the major influential figures.
Alternatively, again from a historic perspective, is the more academic, The Geonim of Babylonia and the Shaping of Medieval Jewish Culture. The blurb there read as follows:
The Geonic period from about the late sixth to mid-eleventh centuries is of crucial importance in the history of Judaism. The Geonim, for whom this era is named, were the heads of the ancient talmudic academies of Babylonia. They gained ascendancy over the older Palestinian center of Judaism and were recognized as the leading religious and spiritual authorities by most of the world’s Jewish population. The Geonim and their circles enshrined the Babylonian Talmud as the central canonical work of rabbinic literature and the leading guide to religious practice, and it was a predominantly Babylonian version of Judaism that was transplanted to newer centers of Judaism in North Africa and Europe. Robert Brody’s book―the first survey in English of the Geonic period in almost a century―focuses on the cultural milieu of the Geonim and on their intellectual and literary creativity.
Brody describes the cultural spheres in which the Geonim were active and the historical and cultural settings within which they functioned. He emphasizes the challenges presented by other Jewish institutions and individuals, ranging from those within the Babylonian Jewish setting―especially the political leadership represented by the Exilarch―to the competing Palestinian Jewish center and to sectarian movements and freethinkers who rejected rabbinic authority altogether. He also describes the variety of ways in which the development of Geonic tradition was affected by the surrounding non-Jewish cultures, both Muslim and Christian
Finally, you might want to try Rabbi Berel Wein's Herald of Destiny - on his website here it lays out the structure of the book and in the first section he spends much time look at the Geonim from their rise till the end of the period.
As far as exploring their responsa, Louis Ginzberg's Geonica is available in its entirety on Hebrewbooks