Since their discovery in the late 1960's the population of known neutron stars has grown to ∼2500. This sample has yielded many surprises and demonstrated that the observational properties of neutron stars are remarkably diverse. The surveys that will be performed with the SKA will produce a further tenfold increase in the number of Galactic neutron stars known. Moreover, the SKA's broad spectral coverage, sub-arraying and multi-beaming capabilities will allow us to characterise these sources with unprecedented efficiency, in turn enabling a giant leap in the understanding of their properties. We review the neutron star population and outline strategies for studying each of the growing number of diverse classes that are populating the "neutron star zoo". Questions that will be addressed by the much larger statistical samples and vastly improved timing efficiency provided by SKA include: (i) the spin period and spin-down rate distributions (and thus magnetic fields) at birth, and the associated information about the supernovae wherein they are formed; (ii) the radio pulsar-magnetar connection; (iii) the link between normal radio pulsars, intermittent pulsars and rotating radio transients; (iv) the slowest possible spin period for a radio pulsar (revealing the conditions at the pulsar death-line); (v) proper motions of pulsars (revealing supernova kick physics); (vi) the mass distribution of neutron stars; (vii) the fastest possible spin period for a recycled pulsar (constraining magnetosphere-accretion disc interactions, gravitational wave radiation and the equation-of-state); (viii) the origin of high eccentricity millisecond pulsars; (ix) the formation channels for recently identified triple systems; and finally (x) how isolated millisecond pulsars are formed. We can also expect that the first phase of the SKA (SKA1), and in particular the full SKA (SKA2), will break new ground unveiling exotic and heretofore unknown systems that will challenge our current knowledge and theories, thus fostering the development of new research areas. Some possibilities for future landmark discoveries representing significant milestones in the astrophysics of compact objects include: (i) sub-millisecond pulsars; (ii) neutron stars born as millisecond pulsars; (iii) neutron stars with masses below 1.1 or above 2.5 M ; (iv) neutron star-black hole binaries; and (v) a triple system containing a pair of neutron stars.Advancing Astrophysics with the Square Kilometre Array