  For the existence of a branched covering Sigma~ --> Sigma between closed
surfaces there are easy necessary conditions in terms of chi(Sigma~),
chi(Sigma), orientability, the total degree, and the local degrees at the
branching points. A classical problem dating back to Hurwitz asks whether these
conditions are also sufficient. Thanks to the work of many authors, the problem
remains open only when Sigma is the sphere, in which case exceptions to
existence are known to occur. In this paper we describe new infinite series of
exceptions, in particular previously unknown exceptions with Sigma~ not the
sphere and with more than three branching points. All our series come with
systematic explanations, based on several different techniques (including
dessins d'enfants and decomposability) that we exploit to attack the problem,
besides Hurwitz's classical technique based on permutations. Using
decomposability we also establish an easy existence result.
