For a topological space $\mathcal X$ and a quotient space $\mathcal Y$ of $\mathcal X$, Wikipedia's page on quotient spaces says that the topological dimension of $\mathcal Y$ can be more than that of $\mathcal X$, for instance in the case of space-filling curves.
Are there conditions under which $\mathcal Y$ is known to be of finite topological dimension, or where $\mathrm{dim} Y <= \mathrm{dim} X$? I am particularly interested in the case when $\mathcal Y = \mathcal X / G$ is the quotient by a group $G$, though any leads would be nice. For motivation, I am interested in when quotient spaces can be embedded into Euclidean space, and it is known that many spaces with finite dimension can be embedded into Euclidean space.