In this lecture we introduce a spacial moduli space $\sf T$ of the pairs formed by definite Calabi-Yau $n$-folds (arising from the Dwork family) along with $n+ 1$ differential $n$-forms. We observe that there exists a unique vector field $\textsf{R}$ on $\sf T$, called modular vector field, satisfying a certain equation involving the Gauss-Manin connection. It turns out that the $q$-expansion (Fourier series) of the components of a solution of $\sf R$, which are called Calabi-Yau modular forms, has integer coefficients, up to multiplying by a constant rational number. In particular, in the case of elliptic curves and $K3$-surfaces, where $n=1,2$, the components of a solution can be written in terms of (quasi-)modular forms satisfying certain enumerative properties. A very useful result of these works is that the modular vector field ${\sf R}$ together with the radial vector field and a degree zero vector field generates a copy of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. We will finish this talk by endowing the space of Calabi-Yau modular forms with an algebraic structure called Rankin-Cohen algebra. To get the vector field $\sf R$ we use an algebraic method called Gauss-Manin connection in disguise introduced by Hossein Movasati.