  Selection of a group of representatives satisfying certain fairness
constraints, is a commonly occurring scenario. Motivated by this, we initiate a
systematic algorithmic study of a \emph{fair} version of \textsc{Hitting Set}.
In the classical \textsc{Hitting Set} problem, the input is a universe
$\mathcal{U}$, a family $\mathcal{F}$ of subsets of $\mathcal{U}$, and a
non-negative integer $k$. The goal is to determine whether there exists a
subset $S \subseteq \mathcal{U}$ of size $k$ that \emph{hits} (i.e.,
intersects) every set in $\mathcal{F}$. Inspired by several recent works, we
formulate a fair version of this problem, as follows. The input additionally
contains a family $\mathcal{B}$ of subsets of $\mathcal{U}$, where each subset
in $\mathcal{B}$ can be thought of as the group of elements of the same
\emph{type}. We want to find a set $S \subseteq \mathcal{U}$ of size $k$ that
(i) hits all sets of $\mathcal{F}$, and (ii) does not contain \emph{too many}
elements of each type. We call this problem \textsc{Fair Hitting Set}, and
chart out its tractability boundary from both classical as well as multivariate
perspective. Our results use a multitude of techniques from parameterized
complexity including classical to advanced tools, such as, methods of
representative sets for matroids, FO model checking, and a generalization of
best known kernels for \textsc{Hitting Set}.
