  We review the physics of purely leptonic decays of $\pi^\pm$, $K^\pm$,
$D^{\pm}$, $D_s^\pm$, and $B^\pm$ pseudoscalar mesons. The measured decay rates
are related to the product of the relevant weak-interaction-based CKM matrix
element of the constituent quarks and a strong interaction parameter related to
the overlap of the quark and antiquark wave-functions in the meson, called the
decay constant $f_P$. The leptonic decay constants for $\pi^\pm$, $K^\pm$,
$D^{\pm}$, $D_s^\pm$, and $B^\pm$ mesons can be obtained with controlled
theoretical uncertainties and high precision from {\it ab initio} lattice-QCD
simulations. The combination of experimental leptonic decay-rate measurements
and theoretical decay-constant calculations enables the determination of
several elements of the CKM matrix within the standard model. These
determinations are competitive with those obtained from semileptonic decays,
and also complementary because they are sensitive to different quark
flavor-changing currents. They can also be used to test the unitarity of the
first and second rows of the CKM matrix. Conversely, taking the CKM elements
predicted by unitarity, one can infer "experimental" values for $f_P$ that can
be compared with theory. These provide tests of lattice-QCD methods, provided
new-physics contributions to leptonic decays are negligible at the current
level of precision. This review is the basis of the article in the Particle
Data Group's 2016 edition, updating the versions in Refs. [1-3].
