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Kolya Uraltsev was one of the inventors of the Heavy Quark Expansion (HQE),that describes inclusive weak decays of hadrons containing heavy quarks and inparticular lifetimes. Besides giving a pedagogic introduction to the subject,we review the development and the current status of the HQE, which justrecently passed several non-trivial experimental tests with an unprecedentedprecision. In view of many new experimental results for lifetimes of heavyhadrons, we also update several theory predictions: $\tau (B^+) / \tau (B_d) =1.04^{+0.05}_{-0.01} \pm 0.02 \pm 0.01$, $\tau (B_s) / \tau (B_d) = 1.001 \pm0.002$, $\tau (\Lambda_b)/ \tau (B_d) = 0.935 \pm 0.054$ and $\bar {\tau}(\Xi_b^0) / \bar{\tau} (\Xi_b^+) = 0.95 \pm 0.06$. The theoretical precision iscurrently strongly limited by the unknown size of the non-perturbative matrixelements of four-quark operators, which could be determined with latticesimulations.

Lifetimes and HQE