Two groups are called commensurable if they have isomorphic subgroups of finite index. In the first part of the talk I will discuss our results with Montse Casals-Ruiz and Ilya Kazachkov on the commensurability classification of right-angled Artin groups (RAAGs) defined by trees. In the second part of the talk I will mention some algorithmic properties of RAAGs and discuss our results with Jordi Delgado and Enric Ventura on the subgroup intersection problem for Droms RAAGs.