  A reparametrization (of a continuous path) is given by a surjective weakly
increasing self-map of the unit interval. We show that the monoid of
reparametrizations (with respect to compositions) can be understood via
``stop-maps'' that allow to investigate compositions and factorizations, and we
compare it to the distributive lattice of countable subsets of the unit
interval. The results obtained are used to analyse the space of traces in a
topological space, i.e., the space of continuous paths up to reparametrization
equivalence. This space is shown to be homeomorphic to the space of regular
paths (without stops) up to increasing reparametrizations. Directed versions of
the results are important in directed homotopy theory.
