  ATLAS measurements of two-particle correlations are presented for $\sqrt{s} =
5.02$ and $13$ TeV $pp$ collisions and for $\sqrt{s_{\mathrm{NN}}} = 5.02$ TeV
$p$+Pb collisions at the LHC. The correlation functions are measured as a
function of relative azimuthal angle $\Delta \phi$, and pseudorapidity
separation $\Delta \eta$, using charged particles detected within the
pseudorapidity interval $|\eta|{<}2.5$. Azimuthal modulation in the long-range
component of the correlation function, with $|\Delta\eta|{>}2$, is studied
using a template fitting procedure to remove a "back-to-back" contribution to
the correlation function that primarily arises from hard-scattering processes.
In addition to the elliptic, $\cos{(2\Delta\phi)}$, modulation observed in a
previous measurement, the $pp$ correlation functions exhibit significant
$\cos{(3\Delta\phi)}$ and $\cos{(4\Delta\phi)}$ modulation. The Fourier
coefficients $v_{n,n}$ associated with the $\cos{(n\Delta\phi)}$ modulation of
the correlation functions for $n =$ $2$-$4$ are measured as a function of
charged-particle multiplicity and charged-particle transverse momentum. The
Fourier coefficients are observed to be compatible with $\cos{(n\phi)}$
modulation of per-event single-particle azimuthal angle distributions. The
single-particle Fourier coefficients $v_n$ are measured as a function of
charged-particle multiplicity, and charged-particle transverse momentum for $n
{=} $ $2$-$4$. The integrated luminosities used in this analysis are, $64$
$\mathrm{nb^{-1}}$ for the $\sqrt{s}=13$ TeV $pp$ data, $170$
$\mathrm{nb^{-1}}$ for the $\sqrt{s}=5.02$ TeV $pp$ data and $28$
$\mathrm{nb^{-1}}$ for the $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV $p$+Pb data.
