When calculus was originally developed, it made heavy use of the notion of an infinitesimal - a quantity larger than zero, but smaller than any positive real number. As George Berkeley famously criticised them, "they are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?" Of course, rather embarrassingly for early analysts, infinitesimals don't actually exist... or do they? This talk will use some ideas from model theory to construct a number system called the hyperreals, where infinitesimals do exist and can be used to do calculus in a more straightforward, intuitive way than the usual approach via limits. No knowledge of model theory (or anything else!) will be assumed.