Recently, there has been a surge in new results on the digital properties of numbers. In this talk, we discuss some recent results on palindromes, numbers which are the same backwards and forwards (e.g. 12321 or 7887) in a fixed base. Despite being elementary objects of study, in order to prove deep results about their multiplicative properties we need to turn to powerful estimates on exponential sums. For this talk, we focus particularly on a recent result of Bryce Kerr and myself, establishing the infinitude of square-free palindromes.