While much of network design focuses mostly on cost (number or weight of edges), node degrees have also played an important role. They have traditionally either appeared as an objective, to minimize the maximum degree (e.g., the Minimum Degree Spanning Tree problem), or as constraints which might be violated to give bicriteria approximations (e.g., the Minimum Cost Degree Bounded Spanning Tree problem). We extend the study of degrees in network design in two ways. First, we introduce and study a new variant of the Survivable Network Design Problem where in addition to the traditional objective of minimizing the cost of the chosen edges, we add a constraint that the $\ell_p$-norm of the node degree vector is bounded by an input parameter. This interpolates between the classical settings of maximum degree (the $\ell_{\infty}$-norm) and the number of edges (the $\ell_1$-degree), and has natural applications in distributed systems and VLSI design. We give a constant bicriteria approximation in both measures using convex programming. Second, we provide a polylogrithmic bicriteria approximation for the Degree Bounded Group Steiner problem on bounded treewidth graphs, solving an open problem from [Kortsarz and Nutov, Discret. Appl. Math. 2022] and [Guo et al., Algorithmica 2022].