  We prove several reflection theorems on $D$-spaces, which are Hausdorff
topological spaces $X$ in which for every open neighbourhood assignment $U$
there is a closed discrete subspace $D$ such that \[ \bigcup\{U(x): x\in D\}=X.
\] The upwards reflection theorems are obtained in the presence of a forcing
axiom, while most of the downwards reflection results use large cardinal
assumptions.
  The combinatorial content of arguments showing that a given space is a
$D$-space, can be formulated using the concept of discrete families. We note
the connection between non-reflection arguments involving discrete families and
the well known question of the existence of families allowing partial
transversals without having a transversal themselves, and use it to give
non-trivial instances of the incompactness phenomenon in the context of
discretisations.
