This thesis is constituted by two independent parts. In the first part we discuss several problems which relate to the Analyst's Travelling Salesman Theorem of Peter Jones and the theories of uniformly rectifiable sets and quasiminimal sets of David and Semmes. To say more, this first part splits naturally into two chapters, which are nevertheless connected. In a first chapter, we present a joint work with Jonas Azzam which shows that, in some sense, there exists a travelling salesman theorem for each property that characterises uniformly rectifiable sets. In a second chapter, we address the question of what type of geometric object can play the role of a curve in a higher dimensional setting, at least from the point of view of an Analyst's TST. The second part of this thesis is dedicated to presenting a joint work with Benjamin Jaye and Xavier Tolsa which settles a longstanding conjecture of Carleson, known in the field as Carleson ve² conjecture, in a positive sense.