The main object of this talk are one-dimensional subshifts, that is, sets of two-sided infinite strings of symbols. Sometimes, finitely many of their elements already determine important properties of the subshift, like certain asymptotics and which finite strings of symbols occur. These elements are called the leading sequences. The notion stems from joint work with R. Grigorchuk, D. Lenz and T. Nagnibeda, where it was introduced to ensure uniform convergence of so-called cocycles. We briefly discuss this original motivation, but focus on the implications of leading sequences for fundamental properties of the subshift like minimality and ergodicity.