For a smooth manifold $M$, the de Rham complex is $\Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \Omega^2(M) \xrightarrow{d} \cdots$ where $\Omega^k(M)$ is the space of smooth $k$-forms and $d$ is the exterior derivative. The condition $d^2 = 0$ is the Poincare lemma.

For a topological space $X$, the singular chain complex has $C_n(X) = \mathbb{Z}[\mathrm{Sing}_n(X)]$ (free abelian group on singular $n$-simplices) with the boundary map $\partial_n = \sum_{i=0}^n (-1)^i \partial_i$. The condition $\partial^2 = 0$ follows from the alternating signs.

For a commutative ring $R$ and elements $x_1, \ldots, x_n \in R$, the Koszul complex $K(x_1, \ldots, x_n)$ is defined on the exterior algebra $\bigwedge^p R^n$ with the differential $d(e_{i_1} \wedge \cdots \wedge e_{i_p}) = \sum_j (-1)^{j+1} x_{i_j} \, e_{i_1} \wedge \cdots \wedge \hat{e}_{i_j} \wedge \cdots \wedge e_{i_p}$.

Any morphism $f : A \to B$ in an abelian category gives a complex $\cdots \to 0 \to A \xrightarrow{f} B \to 0 \to \cdots$ concentrated in two degrees. The cohomology is $\ker f$ and $\mathrm{coker}\, f$.

Any object $A \in \mathcal{A}$ gives a complex $A[0]$ concentrated in degree $0$: all other terms are zero and all differentials are zero. This gives an embedding $\mathcal{A} \hookrightarrow \mathbf{Ch}(\mathcal{A})$.

For a sheaf $\mathcal{F}$ and an open cover $\mathcal{U} = \{U_i\}$ of $X$, the Cech complex $\check{C}^\bullet(\mathcal{U}, \mathcal{F})$ has $\check{C}^p = \prod_{i_0 < \cdots < i_p} \mathcal{F}(U_{i_0} \cap \cdots \cap U_{i_p})$ with the alternating restriction maps as differentials.

For a group $G$ and a $G$-module $M$, the bar resolution $\cdots \to \mathbb{Z}[G^3] \to \mathbb{Z}[G^2] \to \mathbb{Z}[G] \to \mathbb{Z} \to 0$ is a free resolution of $\mathbb{Z}$ as a $G$-module. Applying $\mathrm{Hom}_G(-, M)$ gives the complex computing group cohomology $H^n(G, M)$.

For a chain map $f : A^\bullet \to B^\bullet$, the mapping cone $\mathrm{Cone}(f)$ is the complex with $\mathrm{Cone}(f)^n = A^{n+1} \oplus B^n$ and differential $d(a, b) = (-d_A(a), f(a) + d_B(b))$. The condition $d^2 = 0$ uses $f$ being a chain map.

A double complex $A^{p,q}$ has horizontal differentials $d_h : A^{p,q} \to A^{p+1,q}$ and vertical differentials $d_v : A^{p,q} \to A^{p,q+1}$ with $d_h^2 = 0$, $d_v^2 = 0$, and $d_h \circ d_v + d_v \circ d_h = 0$ (anticommutativity). The total complex $\mathrm{Tot}(A)^n = \bigoplus_{p+q=n} A^{p,q}$ is a single complex.

A complex of sheaves $\mathcal{F}^\bullet$ on $X$ is a complex in the abelian category $\mathbf{Sh}(X)$. The hypercohomology $\mathbb{H}^n(X, \mathcal{F}^\bullet)$ generalizes ordinary sheaf cohomology.

A projective resolution $\cdots \to P_2 \to P_1 \to P_0 \to M \to 0$ of a module $M$ gives a complex $P_\bullet$ with $P_\bullet \to M[0]$ a quasi-isomorphism. This is the starting point for computing derived functors.

The shift $A[k]$ of a complex $A^\bullet$ is defined by $A[k]^n = A^{n+k}$ with differential $d_{A[k]}^n = (-1)^k d_A^{n+k}$. The shift is an autoequivalence of $\mathbf{Ch}(\mathcal{A})$ and plays a central role in triangulated categories.