  We study the set $\operatorname{MA}(X,Y)$ of operators between Banach spaces
$X$ and $Y$ that attain their minimum norm, and the set
$\operatorname{QMA}(X,Y)$ of operators that quasi attain their minimum norm. We
characterize the Radon-Nikodym property in terms of operators that attain their
minimum norm and obtain some related results about the density of the sets
$\operatorname{MA}(X,Y)$ and $\operatorname{QMA}(X,Y)$. We show that every
infinite-dimensional Banach space $X$ has an isomorphic space $Y$ such that not
every operator from $X$ to $Y$ quasi attains its minimum norm. We introduce and
study Bishop-Phelps-Bollob\'as type properties for the minimum norm, including
the ones already considered in the literature, and we exhibit a wide variety of
results and examples, as well as exploring the relations between them.
