We will review some general concepts of deformation theory. Then we will apply these ideas in order to explore the geometry of the moduli space Inv of foliations on a given variety $X$ around the points corresponding to foliations induced by Lie group actions. More precisely, let $X$ be a smooth projective variety over the complex numbers and $S(d)$ the scheme parametrizing $d$-dimensional Lie subalgebras of $H^0(X,\mathcal{T} X)$. For every $\mathfrak{g} \in S(d)$ one can consider the corresponding element $\mathcal{F}(\mathfrak{g})\in Inv$, whose generic leaf coincides with an orbit of the action of $\exp(\mathfrak{g})$ on $X$. We will show that under mild hypotheses, after taking a stratification $\coprod_i S(d)_i\to S(d)$ this assignment yields an isomorphism $\coprod_i S(d)_i\to Inv$ locally around $\mathfrak{g}$ and $\mathcal{F}(\mathfrak{g})$.