Josephus was a Jewish and Roman historian in the 1st Century AD who wrote (amongst other things) about the Roman-Jewish war that lead to the destruction of the Temple in Jerusalem. In the 18th Century this book was widely read by Christians as it appears to provide historical evidence for Jesus; and Josephus was held up as one of the great historians. However to Jews he was a much more controversial figure and wasn't read or referred to until much later in the Enlightenment.
The Talmud is one of the most significant texts in Judaism, second in importance only to the Torah. It is in part a commentary on the Torah, and in part an ongoing discussion (or argument) between various Rabbis & sages about Judaism, the Law and how to interpret the Law. The three experts who discussed the Talmud on In Our Time were Philip Alexander (University of Manchester), Rabbi Norman Solomon (Oxford Centre for Jewish and Hebrew Studies) and Laliv Clenman (Leo Baeck College and King's College London).