  This paper deals with the \emph{Bishop-Phelps-Bollob\'as property}
(\emph{BPBp} for short) on bounded closed convex subsets of a Banach space $X$,
not just on its closed unit ball $B_X$. We firstly prove that the \emph{BPBp}
holds for bounded linear functionals on arbitrary bounded closed convex subsets
of a real Banach space. We show that for all finite dimensional Banach spaces
$X$ and $Y$ the pair $(X,Y)$ has the \emph{BPBp} on every bounded closed convex
subset $D$ of $X$, and also that for a Banach space $Y$ with property $(\beta)$
the pair $(X,Y)$ has the \emph{BPBp} on every bounded closed absolutely convex
subset $D$ of an arbitrary Banach space $X$. For a bounded closed absorbing
convex subset $D$ of $X$ with positive modulus convexity we get that the pair
$(X,Y)$ has the \emph{BPBp} on $D$ for every Banach space $Y$. We further
obtain that for an Asplund space $X$ and for a locally compact Hausdorff $L$,
the pair $(X, C_0(L))$ has the \emph{BPBp} on every bounded closed absolutely
convex subset $D$ of $X$. Finally we study the stability of the \emph{BPBp} on
a bounded closed convex set for the $\ell_1$-sum or $\ell_{\infty}$-sum of a
family of Banach spaces.
