_The Wilderness of the Infinite_ explores the emergence in the Latinate thirteenth century of original approaches to mathematical infinity and to unequal infinities. Within the span of twenty years (1220-1240), Robert Grosseteste and William of Auvergne countenanced the actual infinite and presented very original views on the possibility of comparing infinities. Robert Grosseteste postulated the existence of infinite numbers that measure the number of points in finite line segments. Until his proposal, no one in Western culture had operated with infinite numbers. Grosseteste's proposal led to debates on what criteria one should use when assigning 'sizes' to infinite collections with one-to-one correspondence being proposed as a challenge to the part-whole intuition defended by Grosseteste. But the book is not only about Robert Grosseteste, William of Auvergne, and their impact on medieval philosophy in the period up to 1275. Rather, the historical investigation is instrumental in showing that some of the daring ideas proposed by Grosseteste and William of Auvergne although criticized as naïve, or even incoherent, by twentieth century investigators can be given a perfectly coherent development using some recent mathematical theories, namely non-standard analysis and the theory of numerosities. The book thus offers a methodological proposal on how to engage with the history and the philosophy of mathematical infinity.