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In this paper, we study the capacity dimension of the boundary of $CAT(0)$spaces. We first compare the two metrics on the boundary of a hyperbolic$CAT(0)$ space, i.e., the visual metric and the conical metric, and prove thatthey give the same capacity dimension of the boundary. Then we study thecapacity dimension of the boundary of buildings, which is an important class of$CAT(0)$ spaces. Finally, we give a possible method to prove the finiteness ofthe asymptotic dimension of $CAT(0)$ spaces.

On the capacity dimension of the boundary of CAT(0) spaces