Branch groups are a class of groups which act on rooted trees in a way reminiscent of the action of the whole automorphism group of the tree. Since their introduction by Grigorchuk, branch groups have garnered a lot of attention as a rich source of easy to describe examples of groups with exotic properties, such as finitely generated infinite torsion groups and groups of intermediate growth. Despite intensive research, many aspects of these groups remain mysterious, notably the question of which branch groups admit maximal subgroups of infinite index. We will discuss what is known about this question and present new examples of branch groups with maximal subgroups of infinite index, the first among groups of intermediate growth. This is joint work with Alejandra Garrido.