  We introduce two Bishop-Phelps-Bollob\'as moduli which measure, for a given
Banach space, what is the best possible Bishop-Phelps-Bollob\'as theorem in
this space. We show that there is a common upper bound for these moduli for all
Banach spaces and we present an example showing that this bound is sharp. We
prove the continuity of these moduli and an inequality with respect to duality.
We calculate the two moduli for Hilbert spaces and also present many examples
for which the moduli have the maximum possible value (among them, there are
$C(K)$ spaces and $L_1(\mu)$ spaces). Finally, we show that if a Banach space
has the maximum possible value of any of the moduli, then it contains almost
isometric copies of the real space $\ell_\infty^{(2)}$ and present an example
showing that this condition is not sufficient.
