Constructs the Bruhat--Tits building of $G$ and the parahoric subgroups of $G(k)$, using as an input the apartments constructed in the previous chapter. The key technical device here is that of a \emph{concave function}. This is a function $f : \hat\Phi \to \R$, where $\Phi$ is the root system of $G$ relative to a maximal $k$-split torus $S \subset G$ and $\hat\Phi=\Phi \cup \{0\}$, satisfying the concavity property $f(a+b) \leq f(a)+f(b)$ for all $a,b \in \hat\Phi$ such that $a+b \in \hat\Phi$. If $\cA$ is the apartment associated to $S$ in the previous chapter and $x \in \cA$, the discussion of the previous chapter allows one to construct an open bounded subgroup $G(k)_{x,\,f}$ of $G(k)$. Using the group-theoretic properties of the groups $G(k)_{x,\,f}$ it is shown in \S\ref{sec:iwahori-tits-system} that parahoric subgroups lead to a Tits system, and hence to a (restricted) Tits building. This is the Bruhat--Tits building $\cB(G/k)$ of $G$.