In the vast blackness of space, our home planet is a single sparkling oasis of life. Whether the Universe harbors other worlds that can support even simple life is a question that has been pondered, yet remained unanswered, for over two thousand years. Motivated by the discoveries over the past decade of hundreds of extrasolar planets, NASA and ESA have initiated a series of mission concept studies for space-based astronomical observatories that will find and study extrasolar terrestrial planets, searching for signs of life. This chapter reviews our current understanding of how we will identify planets that might support life around other stars, and incorporates results from the new science of astrobiology that support these searches