Measurements of two- and multi-particle angular correlations in pp collisions at s=5,7, and 13TeV are presented as a function of charged-particle multiplicity. The data, corresponding to integrated luminosities of 1.0pb −1 (5 TeV), 6.2pb −1 (7 TeV), and 0.7pb −1 (13 TeV), were collected using the CMS detector at the LHC. The second-order (v 2 ) and third-order (v 3 ) azimuthal anisotropy harmonics of unidentified charged particles, as well as v 2 of K S 0 and Λ/Λ‾ particles, are extracted from long-range two-particle correlations as functions of particle multiplicity and transverse momentum. For high-multiplicity pp events, a mass ordering is observed for the v 2 values of charged hadrons (mostly pions), K S 0 , and Λ/Λ‾, with lighter particle species exhibiting a stronger azimuthal anisotropy signal below p T ≈2GeV/c. For 13 TeV data, the v 2 signals are also extracted from four- and six-particle correlations for the first time in pp collisions, with comparable magnitude to those from two-particle correlations. These observations are similar to those seen in pPb and PbPb collisions, and support the interpretation of a collective origin for the observed long-range correlations in high-multiplicity pp collisions.