  Brownian motion is a universal characteristic of colloidal particles embedded
in a host medium, and it is the fingerprint of molecular transport or
diffusion, a generic feature of relevance not only in Physics but also in
several branches of Science and Engineering. Since its discovery, Brownian
motion or colloid dynamics has been important in elucidating the connection
between the molecular details of the diffusing macromolecule and the
macroscopic information of the host medium. However, colloid dynamics is far
from being completely understood. For example, the diffusion of non-spherical
colloids and the effects of geometry on the dynamics of either passive or
active colloids are a few representative cases that are part of the current
challenges in Soft Matter Physics. In this contribution, we take a step forward
to introduce a covariant description of the colloid dynamics in curved spaces.
This formalism will allow us to understand several phenomena, for instance, the
effects of curvature on the kinetics during spinodal decomposition and the
thermodynamic properties of the colloidal dispersion, just to mention a few
examples. This theoretical framework will also serve as the starting point to
highlight the role of geometry on colloid dynamics, an aspect that is of
paramount importance to understanding more complex phenomena, such as the
diffusive mechanisms of proteins embedded in cell membranes.
