  The known duality of the space of Bloch complex-valued functions on the open
complex unit disc $\mathbb{D}$ is addressed under a new approach with the
introduction of the concepts of Bloch molecules and Bloch-free Banach space of
$\mathbb{D}$. We introduce the notion of compact Bloch mapping from
$\mathbb{D}$ to a complex Banach space and establish its main properties:
invariance by M\"obius transformations, linearization from the Bloch-free
Banach space of $\mathbb{D}$, factorization of their derivatives, inclusion
properties, Banach ideal property and transposition on the Bloch function
space. We state Bloch versions of the classical theorems of Schauder,
Gantmacher and Davis-Figiel-Johnson-Pelczy\'nski.
