  We extend the concept of orbifold to that of branchfold, in order to allow
any cone singularities with rational angles, and show why branchfolds naturally
fit in the theory of branched coverings. Then, we obtain a geometric goodness
theorem for branchfolds and apply it to prove that a conifold can be endowed
with branchfold structure if and only if it has locally finite holonomy.
