In this paper, we introduce and study the idea of a new type of compactness, defined in terms of a grill $\cal G$ in a topological space $X$. Calling it $\cal G$-compactness, we investigate its relation with compactness, among other things. Analogues of Alexender's subbase theorem and Tychonoff product theorem are also obtained for $\cal G$-compactness. Finally, we exhibit a new method, in terms of the deliberations here, for construction of the well known one-point compactification of a $T_2$, locally compact and noncompact topological space.