Did the Indian and Babylonian astronomy evolve in isolation, was there mutual influence, or was one dependent on the other? Scholars have debated these questions for more than two centuries, and opinion has swung one way or the other with time. The similarities between the two systems that have been investigated are: the use of 30 divisions of the lunar month; the 360 divisions of the civil year; the length of the year; and the solar zodiac. Some have wondered if the Babylonian planetary tables might have played a role in the theories of the siddhantas. I shall in this essay go over the essentials of the early Indian and Babylonian astronomy and summarize the latest views on the relationship between them. I shall show that the key ideas found in the Babylonian astronomy of 700 BC are already present in the Vedic texts, which even by the most conservative reckoning are centuries older than this period. I shall also show that the solar zodiac (rashis) was used in Vedic India and I shall present a plausible derivation of the symbols of the solar zodiac from the deities of the lunar segments. This does not mean that the Babylonian astronomy and astrology is derived from the Indian tradition. If at all there was borrowing, that was restricted to the most general ideas only. The nature of Indian and Babylonian astronomical methods is quite different. I propose that it is most likely that Babylonian astronomy emerged independently.