The Mayer-Vietoris sequence is a powerful tool to compute the integer coefficients homology of a topological space. It is a long exact sequence relating the homology of a topological space, the homologies of covering subsets and the homologies of their intersections.
Let $X$ be a topological space and let $A$ and $B$ be two subsets of $X$ whose interiors cover $X$, then there exists a long exact sequence $$\cdots\xrightarrow[]{}H_{n+1}(X;\mathbb{Z})\xrightarrow[]{\partial_n}H_n(A\cap B;\mathbb{Z})\xrightarrow[]{(i_*,j_*)} H_n(A;\mathbb{Z})\oplus H_n(B;\mathbb{Z})\xrightarrow[]{k_*-\ell_*} H_n(X;\mathbb{Z})\xrightarrow[]{}\cdots,$$ where $i$, $j$, $k$ and $\ell$ are the inclusions of $A\cap B$ in $A$, $A\cap B$ in $B$, $A$ in $X$ and $B$ in $X$, respectively.
It is a homological version of the Seifert-van Kampen theorem for the fundamental group.