In several previous chapters, we have been primarily concerned with providing upper bounds on the worst-case risk of specific procedures. Here, we seek instead lower bounds on the worst-case risk of any procedure. Such minimax lower bounds provide a benchmark against which we can measure the performance of proposed methods. The three main types of minimax lower bound we consider arise from Le Cam's two-point lemma, Assouad's lemma and Fano's lemma, and we will see several examples of how these results can be applied. Finally, we establish tools for controlling minimax quantiles, in an attempt to avoid the loss of information in reducing the (random) loss to its expectation (risk). In many cases, it is possible to characterise the dependence of minimax quantiles on the quantile level, thereby providing a finer-grained understanding of the difficulty of a statistical problem.