We present measurements of the elliptic (ser=111111111&_pii=S0370269316303586&_rdoc=1&_issn=03702693&md5=9ca06101309bad52b1ff201bcbd21958" title="Click to view the MathML source">v2), triangular (ser=111111111&_pii=S0370269316303586&_rdoc=1&_issn=03702693&md5=4285627d45ccdda4c0d45ff7ed4a256b" title="Click to view the MathML source">v3) and quadrangular (ser=111111111&_pii=S0370269316303586&_rdoc=1&_issn=03702693&md5=afabd775fa7380dd028f231d672a093d" title="Click to view the MathML source">v4) anisotropic azimuthal flow over a wide range of pseudorapidities (ser=111111111&_pii=S0370269316303586&_rdoc=1&_issn=03702693&md5=8a85b44dae8fb199e19945be549092d8" title="Click to view the MathML source">−3.5<η<5). The measurements are performed with Pb–Pb collisions at ser=111111111&_pii=S0370269316303586&_rdoc=1&_issn=03702693&md5=d8ce3f723832d73f2f75919ffd39233c"> using the ALICE detector at the Large Hadron Collider (LHC). The flow harmonics are obtained using two- and four-particle correlations from nine different centrality intervals covering central to peripheral collisions. We find that the shape of ser=111111111&_pii=S0370269316303586&_rdoc=1&_issn=03702693&md5=75251f1f46549c175e779ff2b4ae8a8d" title="Click to view the MathML source">vn(η) is largely independent of centrality for the flow harmonics ser=111111111&_pii=S0370269316303586&_rdoc=1&_issn=03702693&md5=ea9a291dd8beedf88a13c9103c79334a" title="Click to view the MathML source">n=2–4, however the higher harmonics fall off more steeply with increasing ser=111111111&_pii=S0370269316303586&_rdoc=1&_issn=03702693&md5=8f072ec63cf8a21187020234d0a227ae" title="Click to view the MathML source">|η|. We assess the validity of extended longitudinal scaling of ser=111111111&_pii=S0370269316303586&_rdoc=1&_issn=03702693&md5=9ca06101309bad52b1ff201bcbd21958" title="Click to view the MathML source">v2 by comparing to lower energy measurements, and find that the higher harmonic flow coefficients are proportional to the charged particle densities at larger pseudorapidities. Finally, we compare our measurements to both hydrodynamical and transport models, and find they both have challenges when it comes to describing our data.