The Riemann-Hurwitz formula establishes a relation between the degree and the number of critical points of branched coverings $f:X\to Y$. This relation involves the Euler characteristic of the spaces $X$ and $Y$. A priori, it has no sense when the spaces are not locally connected.The formula holds for branched coverings between finite cell complexes and also between open connected subsets of the sphere. We use it to define the Euler characteristic of a continuum even if it is not locally connected. For example, $\chi (X)=0$ for a solenoid and $\chi(K)=1/2$, where $K$ is the Knaster continuum. We give applications to dynamics of sphere branched coverings and provide several examples illustrating the interest of the results obtained.