  In the last years self-motile droplets attracted the attention of scientists
from different fields ranging from applied biology to theoretical physics,
because of their promising technological applications and important biological
implications. In this Chapter we review the state of the art of the research on
active droplets with a particular focus on theoretical and numerical studies.
In particular, we reviewed the active gel theory, namely a generalization of
the standard Landau-de Gennes theory for liquid crystals adapted to take into
account internal active injection due to the presence of self-motile
constituents. When confined in finite geometries, liquid crystalline-like
systems are also subject to topological constraints. Because of the relevance
of topology in many different realizations of active droplets, we also reviewed
some fundamental topological concepts. We review how motility arises in
different realizations of active droplet both in 2d and 3d as the result of the
breaking of specific symmetries, by looking in particular detail at the case of
polar and nematic droplets and shells of active liquid crystals.
