Sub-Riemannian and sub-Finsler Carnot groups emerge as limit metric spaces, both as distinguished asymptotic spaces and as tangent metric spaces. In this chapter, we begin by reviewing the notion of limits of metric spaces and consequently introduce the concept of asymptotic cones and metric tangent spaces; see Sect. 12.1. In Sect. 12.2, to study varying CC structure, we discuss some terminology and some preliminary results, which mostly are consequences of Grönwall Lemma. Section 12.3 is a collection of examples, which help the understanding of the more general statements and proofs contained in the subsequent chapters. In Sect. 12.4, we explain and prove Pansu’s Theorem on the asymptotic geometry of nilpotent Lie groups. In Sect. 12.5, we discuss Mitchell’s Theorem: we give a complete proof in the case of sub-Finsler Lie groups, showing that the tangent spaces are Carnot groups. We only mention the general result for sub-Finsler manifolds in Sect. 12.6, explaining what is the general strategy of proof for varying CC structures; see Sect. 12.7. We conclude the chapter with Sect. 12.8, where we discuss finitely-generated groups of polynomial growth and we mention the general strategy to prove a celebrated result of Gromov proving that these groups are virtually nilpotent and hence their asymptotic cones are sub-Finlser Carnot groups.