I consider ways in which the concept of the infinite has been applied in science and technology. I begin by considering the invention of the calculus, which makes precise some of the ideas that people have had when trying to reckon with the infinitely small and without which science and technology these days would be literally unthinkable. I explain some of the basic ideas of the calculus, and I discuss the dispute between the two seventeenth-century mathematicians Isaac Newton and Gottfried Leibniz concerning which of them could claim the credit for its invention. I also discuss the disagreements that dogged the calculus’s early days, in which the church too got involved. I then turn attention from the infinitely small to the infinitely big. Do space and time go on for ever? Are there infinitely many stars? I look at the various considerations that have been advanced over the centuries for answering these questions, culminating in the contemporary orthodoxy that space is finite, albeit unbounded, and that time too may be finite – with many cosmologists arguing, by appeal to the expansion of the universe, that everything started with a big bang and may all end with a big crunch. I show that scientists are generally wary of the infinite, just as the ancient Greeks were, and that whenever it appears in any of their calculations this forces them back to the drawing board to reconsider what has led them there.