The nuclear dimension of a C*-algebra, introduced by Winter and Zacharias, is a natural generalization to the non-commutative setting of the covering dimension of a topological space. The decomposition rank, due to Kirchberg and Winter, is a slight variation on the nuclear dimension that bounds it from above. In my talk, I will define these two notions of dimension for C*-algebras and explain their relevance to the study and classification of nuclear C*-algebras. I will go over various structural properties enjoyed by the C*-algebras of finite decomposition rank and nuclear dimension. Finally, I will discuss a number of open questions and my work in progress on some of them.