  If $\eta$ is a contact form on a manifold $M$ such that the orbits of the
Reeb vector field form a simple foliation $\mathcal{F}$ on $M$, then the
presymplectic 2-form $d\eta$ on $M$ induces a symplectic structure $\omega$ on
the quotient manifold $N=M/\mathcal{F}$. We call $(M,\eta)$ a $\textit
contactification$ of the symplectic manifold $(N,\omega)$. First, we present an
explicit geometric construction of contactifications of some coadjoint orbits
of connected Lie groups. Our construction is a far going generalization of the
well-known contactification of the complex projective space
$\mathbb{C}P^{n-1}$, being the unit sphere $S^{2n-1}$ in $\mathbb{C}^{n}$, and
equipped with the restriction of the Liouville 1-form on $\mathbb{C}^n$.
Second, we describe a constructive procedure for obtaining contactification in
the process of the Marsden-Weinstein-Meyer symplectic reduction and indicate
geometric obstructions for the existence of compact contactifications. Third,
we show that contactifications provide a nice geometrical tool for a Lagrangian
description of Hamiltonian systems on compact symplectic manifolds
$(N,\omega)$, on which symplectic forms never admit a `vector potential'.
