For $X = U \cup V$ open, the short exact sequence $0 \to C^\bullet(X) \to C^\bullet(U) \oplus C^\bullet(V) \to C^\bullet(U \cap V) \to 0$ gives the Mayer-Vietoris long exact sequence.

For $A \subseteq X$, the sequence $0 \to C^\bullet(A) \to C^\bullet(X) \to C^\bullet(X, A) \to 0$ gives $\cdots \to H^n(A) \to H^n(X) \to H^n(X, A) \xrightarrow{\delta} H^{n+1}(A) \to \cdots$.

From $0 \to M' \to M \to M'' \to 0$ in $R\text{-}\mathbf{Mod}$, applying $\mathrm{Hom}(-, N)$ gives $0 \to \mathrm{Hom}(M'', N) \to \mathrm{Hom}(M, N) \to \mathrm{Hom}(M', N) \to \mathrm{Ext}^1(M'', N) \to \cdots$.

From $0 \to M' \to M \to M'' \to 0$, applying $- \otimes N$ gives $\cdots \to \mathrm{Tor}_1(M'', N) \to M' \otimes N \to M \otimes N \to M'' \otimes N \to 0$.

For $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ on $X$: $0 \to \Gamma(\mathcal{F}') \to \Gamma(\mathcal{F}) \to \Gamma(\mathcal{F}'') \xrightarrow{\delta} H^1(X, \mathcal{F}') \to \cdots$.

$0 \to \underline{\mathbb{Z}} \to \mathcal{O}_X \to \mathcal{O}_X^* \to 0$ gives $\cdots \to H^1(X, \mathcal{O}) \to H^1(X, \mathcal{O}^*) \xrightarrow{c_1} H^2(X, \mathbb{Z}) \to H^2(X, \mathcal{O}) \to \cdots$.

For $0 \to M' \to M \to M'' \to 0$ as $G$-modules: $\cdots \to H^n(G, M') \to H^n(G, M) \to H^n(G, M'') \xrightarrow{\delta} H^{n+1}(G, M') \to \cdots$.

From $0 \to \mathbb{Z}/p \to \mathbb{Z}/p^2 \to \mathbb{Z}/p \to 0$, the connecting homomorphism $\beta : H^n(X; \mathbb{Z}/p) \to H^{n+1}(X; \mathbb{Z}/p)$ is the Bockstein homomorphism, used in Steenrod operations.

For an oriented sphere bundle $S^n \to E \to B$, the Gysin sequence $\cdots \to H^k(B) \to H^{k+n}(E) \to H^{k+n}(B) \xrightarrow{e \cup} H^{k+n+1}(B) \to \cdots$ arises from a long exact sequence.

For a fiber bundle $F \to E \to S^n$ with $n \geq 2$: $\cdots \to H^k(E) \to H^k(F) \to H^{k-n+1}(F) \to H^{k+1}(E) \to \cdots$.

Given a morphism of SES of complexes, the connecting homomorphisms commute with the induced maps. This is essential for functoriality of long exact sequences and comparison arguments using the Five Lemma.

In a triangulated category, a distinguished triangle $A \to B \to C \to A[1]$ and a cohomological functor $H$ give a long exact sequence $\cdots \to H^n(A) \to H^n(B) \to H^n(C) \to H^{n+1}(A) \to \cdots$. The LES in cohomology of complexes is the special case where the triangulated category is $K(\mathcal{A})$ or $D(\mathcal{A})$.