On $D(\mathcal{A})$, the standard t-structure has $D^{\leq 0} = \{A^\bullet : H^n(A) = 0 \text{ for } n > 0\}$ and $D^{\geq 0} = \{A^\bullet : H^n(A) = 0 \text{ for } n < 0\}$. The truncation $\tau_{\leq 0}$ replaces $A^0$ by $\ker d^0$. The heart is $\mathcal{A}$ itself.

On $D^b_c(X)$ (constructible sheaves on a stratified space), the perverse t-structure with perversity $p$ shifts the standard t-structure by the dimension of each stratum. The heart is the abelian category of perverse sheaves $\mathrm{Perv}(X)$.

A torsion pair $(\mathcal{T}, \mathcal{F})$ in an abelian category $\mathcal{A}$ (where $\mathrm{Hom}(\mathcal{T}, \mathcal{F}) = 0$ and every object fits in a SES $0 \to T \to A \to F \to 0$) induces a t-structure on $D^b(\mathcal{A})$ by tilting.

A t-structure is non-degenerate if $\bigcap_n \mathcal{T}^{\leq n} = 0 = \bigcap_n \mathcal{T}^{\geq n}$. The standard t-structure on $D^b(\mathcal{A})$ is non-degenerate. Non-degenerate t-structures allow the recovery of an object from its "cohomology objects."

A weight structure (or co-t-structure) on $\mathcal{T}$ is a t-structure on $\mathcal{T}^{\mathrm{op}}$. Weight structures arise in mixed motives and provide weight filtrations on cohomology.

For a tilting module $T$ over an algebra $A$, with $B = \mathrm{End}(T)$, the derived equivalence $D^b(A) \simeq D^b(B)$ transports the standard t-structure on $D^b(B)$ to a non-standard t-structure on $D^b(A)$.

The cohomology functor $H^0 : D(\mathcal{A}) \to \mathcal{A}$ factors as $\tau_{\geq 0} \circ \tau_{\leq 0}$ and takes values in the heart. More generally, $H^n = H^0 \circ [-n]$ gives the $n$-th cohomology with respect to the t-structure.

A Bridgeland stability condition on $D^b(\mathrm{Coh}(X))$ consists of a t-structure (with heart $\mathcal{A}$) together with a central charge $Z : K_0(\mathcal{A}) \to \mathbb{C}$ satisfying stability conditions. The space of stability conditions is a complex manifold.

For a closed subvariety $Z \subseteq X$ with complement $U$, there is a recollement $D^b_Z(X) \leftrightarrows D^b(X) \leftrightarrows D^b(U)$ that is compatible with t-structures. The perverse t-structure on $D^b_c(X)$ is constructed by gluing t-structures via recollement.

The Beilinson-Bernstein-Deligne (BBD) decomposition theorem states that for a proper map $f : X \to Y$, $Rf_* \mathrm{IC}_X \cong \bigoplus_i {}^p H^i(Rf_* \mathrm{IC}_X)[-i]$ in $D^b_c(Y)$. This uses the perverse t-structure essentially.

For a bounded t-structure on $D^b(\mathcal{A})$ with heart $\mathcal{B}$, there is a realization functor $D^b(\mathcal{B}) \to D^b(\mathcal{A})$. It is not always an equivalence, but it is for the standard t-structure.

Given a locally closed subset $j : U \hookrightarrow X$, the intermediate extension $j_{!*}$ sends perverse sheaves on $U$ to perverse sheaves on $X$. It is defined using the perverse truncation functors and is neither left nor right exact in the classical sense.