Thermal x-rays from neutron stars are mainly radiated by accretion discs originating in the flux of material from a companion star. The companions are white dwarf stars with a range of masses, and some black holes. X-ray bursts are attributed to catastrophic nuclear events on the neutron star surface following accretion from the companion. Structure in the rotating accretion disc is observed as quasi-periodic oscillations (QPOs).

Most of our understanding of the location and nature of the beamed emission comes from the pulse profiles, which are available over the whole electromagnetic spectrum. The radio profiles are the most detailed, with observations of polarisation, width and components.

Finding the population of pulsars in the Milky Way galaxy requires a knowledge of the parameters and limitations of the various surveys made with different instruments and in different regions of the sky. We list the available survey data and show how models of the galactic population can be compared with the observational data, allowing estimates of pulsar birthrate and lifetime. Determination of accurate positions of individual pulsars require a Solar System ephemeris and a complex geometrical computation. Binary pulsar orbits display reletivistic effects which can be measured with remarkable precision to yield parameters of orbits and checks on relativistic theory.

The characteristic steps in the rotation rates of pulsars are known as glitches and arise in the irregular transfer of angular momentum from the interior to the crust as a neutron star spins down. They are related to the structure and the fluid dynamics of some superfluid components. The angular momentum is quantised in vortices, which may be pinned to the crystal structure of the crust. Glitches may be related to catastrophic unpinning events and to cracking of the crust itself. Timing noise is quasi-random variation in rotation rate. In many pulsars, the spin-down rate is seen to switch abruptly as the emission changes, indicating that changes in magnetospheric particle flows are responsible for both spin-down and radiation.

Pulsar distances are obtained from their frequency dispersion, geometrically from annual parallax, and from optical identifiction with supernova remnants, globular clusters and binary companions. For most pulsars, distances are only available from observation of effects of propagation in the interstellar medium, particularly neutral hydrogen absorption and frequency dispersion. Interpretation of the dispersion measure requires a model of the electron distribution through the Galaxy.

Magnetars were originally observed as high-energy emitters as either soft gamma-ray repeaters (SGRs) or anomalous x-ray pulsars (AXPs). They are very active, mainly observed as x-ray sources, apparently very young and probably part of the general population of pulsars but with much larger magnetic fields. The origin of the large magnetic fields is unclear.

Stable neutron stars exist with masses approximately between one and two solar masses, and radii of approximately 10 to 11 km. The structure is determined primarily by a balance between gravitation and the repulsion between adjacent neutrons. The configuration depends on the equation of state of the neutron fluid. The rotation of the strong dipolar magnetic field generates a magnetosphere of charged particles, which co-rotates with the star.

Precision timing of pulses is at the heart of pulsar research. Pulse arrival times can be measured to an accuracy of only a few metres travel time, and analysis must take account of pulsar positions and the Earth’s orbit, the Römer correction to the barycentre, and General Relativistic corrections. Pulsar timing contributes to the comparison of fundamental positional reference frames. Timing provides periods and period changes on short and long time scales, giving pulsar ages and proper motions. The precision timing of some millisecond pulsars is comparable to the best terrestrial laboratory clocks.

Digitisation of incoming signals at nanosecond intervals allows complex manipulation of radio signals to provide for simultaneous multi-beam and multi-frequency operation. The periodic signals from pulsars must be extracted from background noise, allowing for frequency dispersion in propagation through the interstellar medium.

The remnant of a supernova explosion may be observed for some thousands of years in close relation to a pulsar. Radiation from a pulsar may excite radiation from the interstellar medium, causing a pulsar wind nebula, which may be asymmetric due to velocity of the pulsar

The radio and high-energy profiles show that the emitting regions are concentrated in gaps in the magnetosphere located over the magnetic poles and near the velocity of light cylinder. The radio sources of most normal pulsars are distributed unevenly over the polar cap and are highly concentrated, broadband and variable. Their excitation may move laterally, causing drifting in sub-pulse timing. Other radio emitters are located close to the gamma-ray emitters in the outer magnetosphere. Almost all radio pulses are highly polarised; the sweep of position angle in the radio pulses is related to the magnetic field at the location of the emitters.

Despite the extensive knowledge of the characteristics of the coherent radio emission, the mechanism is not understood. The high-energy radiation is incoherent and may be related to the flux of relativistic electrons and positrons in a current sheet at the boundary of the magnetosphere. The radio emission from the polar cap is at lower frequency at larger radii, as the magnetic field lines diverge. The emission may be affected by propagation through the polar cap; refraction along the magnetic field lines may increase the apparent pulse width at lower frequencies.

Advances in observing techniques, the commissioning of new radio telescopes and the prospect of the Square Kilometre Array are opening new fields of pulsar research. The 55 years since the discovery of pulsars have revealed a rich and evolving population and shown how precise timing can transform our understanding of neutron star structure, binary system dynamics, stellar populations and the interstellar medium, and have opened new prospects in general relativity physics. X-ray and gamma-ray telescopes, and Cerenkov shower arrays, are extending observations over the whole electromagnetic spectrum.

The majority of millisecond pulsars are in binary systems with white dwarfs or other neutron stars. Precision timing yields remarkably accurate orbital parameters and their evolution. Binary systems provide tests of relativistic effects including energy loss by gravitational waves.