In this work, we introduce topological representations of a quiver as a system consisting of topological spaces and its relationships determined by the quiver. Such a setting gives a natural connection between topological representations of a quiver and diagrams of topological spaces. Firstly, we investigate the relation between the category of topological representations and that of linear representations of a quiver via $P(\varGamma )$-$\mathcal {TOP}^o$ and $k\varGamma$-Mod, concerning (positively) graded or vertex (positively) graded modules. Secondly, we discuss the homological theory of topological representations of quivers via the $\varGamma$-limit functor $lim ^{\varGamma }$, and use it to define the homology groups of topological representations of quivers via $H _n$. It is found that some properties of a quiver can be read from homology groups. Thirdly, we investigate the homotopy theory of topological representations of quivers. We define the homotopy equivalence between two morphisms in ${\textbf {Top}}\text{- }{} {\textbf {Rep}}\varGamma$ and show that the parallel Homotopy Axiom also holds for top-representations based on the homotopy equivalence. Last, we obtain the functor $At^{\varGamma }$ from ${\textbf {Top}}\text{- }{} {\textbf {Rep}}\varGamma$ to ${\textbf {Top}}$ and show that $At^{\varGamma }$ preserves homotopy equivalence between morphisms. The relationship between the homotopy groups of a top-representation $(T, f)$ and the homotopy groups of $At^{\varGamma }(T,f)$ is also established.