  We study approximation of operators between Banach spaces $X$ and $Y$ that
nearly attain their norms in a given point by operators that attain their norms
at the same point. When such approximations exist, we say that the pair $(X,
Y)$ has the pointwise Bishop-Phelps-Bollob\'as property (pointwise BPB property
for short). In this paper we mostly concentrate on those $X$, called universal
pointwise BPB domain spaces, such that $(X, Y)$ possesses pointwise BPB
property for every $Y$, and on those $Y$, called universal pointwise BPB range
spaces, such that $(X, Y)$ enjoys pointwise BPB property for every uniformly
smooth $X$. We show that every universal pointwise BPB domain space is
uniformly convex and that $L_p(\mu)$ spaces fail to have this property when
$p>2$. For universal pointwise BPB range space, we show that every
simultaneously uniformly convex and uniformly smooth Banach space fails it if
its dimension is greater than one. We also discuss a version of the pointwise
BPB property for compact operators.
