Let $X$ and $Y$ be non-empty topological spaces. The aim of this talk, roughly speaking, to introduce new ways to measure how one can map $X$ to $Y$. More precisely, fundamental group of the pair $(X,Y)$ measures the set of continuous maps $X\times I/(X\times\{0\} \cup X\times  \{1\})$ to the pointed space $(Y,y_0)$, where $I$ is the unit interval and $y_0\in Y$. By fixing $X$, we get a covariant  functor from the category of  pointed topological spaces to the category of groups.  Similarly, if we fix $(Y,y_0)$, then we get a contravariant functor from the category of topological spaces to the category of groups. Various properties of these functors will be discussed. If time permits, I will talk about interrelation groups of a  pair of topological spaces. This is a joint work with Mustafa Topkara and Deniz Kutluay.