ATLAS measurements of two-particle correlations are presented for s=5.02 and 13 TeV pp collisions and for sNN=5.02 TeV p+Pb collisions at the LHC. The correlation functions are measured as a function of relative azimuthal angle Δϕ, and pseudorapidity separation Δη, using charged particles detected within the pseudorapidity interval |η|<2.5. Azimuthal modulation in the long-range component of the correlation function, with |Δη|>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(2Δϕ), modulation observed in a previous measurement, the pp correlation functions exhibit significant cos(3Δϕ) and cos(4Δϕ) modulation. The Fourier coefficients vn,n associated with the cos(nΔϕ) 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(nϕ) modulation of per-event single-particle azimuthal angle distributions. The single-particle Fourier coefficients vn 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 nb−1 for the s=13 TeV pp data, 170 nb−1 for the s=5.02 TeV pp data, and 28 nb−1 for the sNN=5.02 TeV p+Pb data.12 MoreReceived 26 September 2016Revised 13 June 2017DOI:https://doi.org/10.1103/PhysRevC.96.024908Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.©2017 CERN, for the ATLAS CollaborationPhysics Subject Headings (PhySH)Research AreasCollective flowHadron-hadron interactionsParticle correlations & fluctuationsQuantum chromodynamicsQuark-gluon plasmaRelativistic heavy-ion collisionsNuclear Physics