  A minimal family of curves on an embedded surface is defined as a
1-dimensional family of rational curves of minimal degree, which cover the
surface. We classify such minimal families using constructive methods. This
allows us to compute the minimal families of a given surface.
  The classification of minimal families of curves can be reduced to the
classification of minimal families which cover weak Del Pezzo surfaces. We
classify the minimal families of weak Del Pezzo surfaces and present a table
with the number of minimal families of each weak Del Pezzo surface up to Weyl
equivalence.
  As an application of this classification we generalize some results of
Schicho. We classify algebraic surfaces which carry a family of conics. We
determine the minimal lexicographic degree for the parametrization of a surface
which carries at least 2 minimal families.
