Understanding the natal kicks, or birth velocities, of neutron stars is essential for understanding the evolution of massive binaries and double neutron star formation. We use maximum likelihood methods as published in Verbunt et al. to analyse a new large data set of parallaxes and proper motions measured by Deller et al. This sample is roughly three times larger than number of measurements available before. For both the complete sample and its younger part (spin-down ages τ < 3 Myr), we find that a bimodal Maxwellian distribution describes the measured parallaxes and proper motions better than a single Maxwellian with probability of 99.3 and 95.0 per cent, respectively. The bimodal Maxwellian distribution has three parameters: fraction of low-velocity pulsars and distribution parameters σ1 and σ2 for low- and high-velocity modes. For a complete sample, these parameters are as follows: $42_{-15}^{+17}$ per cent, $\sigma _1=128_{-18}^{+22}$ km s−1, and σ2 = 298 ± 28 km s−1. For younger pulsars, which are assumed to represent the natal kick, these parameters are as follows: $20_{-10}^{+11}$ per cent, $\sigma _1=56_{-15}^{+25}$ km s−1, and σ2 = 336 ± 45 km s−1. In the young population, 5 ± 3 per cent of pulsars have velocities less than 60 km s−1. We perform multiple Monte Carlo tests for the method taking into account realistic observational selection. We find that the method reliably estimates all parameters of the natal kick distribution. Results of the velocity analysis are weakly sensitive to the exact values of scale lengths of the Galactic pulsar distribution.

Neutron stars are natural physical laboratories allowing us to study a plethora of phenomena in extreme conditions. In particular, these compact objects can have very strong magnetic fields with non-trivial origin and evolution. In many respects, its magnetic field determines the appearance of a neutron star. Thus, understanding the field properties is important for the interpretation of observational data. Complementing this, observations of diverse kinds of neutron stars enable us to probe parameters of electro-dynamical processes at scales unavailable in terrestrial laboratories. In this review, we first briefly describe theoretical models of the formation and evolution of the magnetic field of neutron stars, paying special attention to field decay processes. Then, we present important observational results related to the field properties of different types of compact objects: magnetars, cooling neutron stars, radio pulsars, and sources in binary systems. After that, we discuss which observations can shed light on the obscure characteristics of neutron star magnetic fields and their behaviour. We end the review with a subjective list of open problems.

Supernova explosion and the associated neutron star (NS) natal kicks are important events on a pathway of a binary to become a gravitational wave source, an X-ray binary, or a millisecond radio pulsar. Weak natal kicks often lead to binary survival, while strong kicks frequently disrupt the binary. In this article, we aim to further constrain NS natal kicks in binaries. We explore binary population synthesis models by varying prescription for natal kick, remnant mass, and mass accretion efficiency. We introduce a robust statistical technique to analyse combined observations of different nature. Using this technique, we further test different models using parallax and proper motion measurements for young isolated radio pulsars and similar measurements for Galactic Be X-ray binaries (BeXs). Our best model for natal kicks is consistent with both measurements and contains a fraction of w = 0.2 ± 0.1 weak natal kicks with |$\sigma _1 = 45^{+25}_{-15}$| km s⁻¹, the remaining natal kicks are drawn from the high-velocity component, same as in previous works: σ₂ = 336 km s⁻¹. We found that currently used models for natal kicks of NSs produced by electron capture supernova (ecSN; combination of Maxwellian σ = 265 km s⁻¹ and σ = 30 km s⁻¹ for electron capture) are inconsistent or marginally consistent with parallaxes and proper motions measured for isolated radio pulsars. We suggest a new model for natal kicks of ecSN, which satisfy both observations of isolated radio pulsars and BeXs.

Initial distributions of pulsar periods and magnetic fields are essential components of multiple modern astrophysical models. Not enough work has been done to properly constrain these distributions using direct measurements. Here, we aim to fill this gap by rigorously analysing the properties of young neutron stars (NSs) associated to supernova remnants (SNRs). In order to perform this task, we compile a catalogue of 56 NSs uniquely paired to SNRs with known age estimate. Further, we analyse this catalogue using multiple statistical techniques. We found that distribution of magnetic fields and periods for radio pulsars are both well described using the lognormal distribution. The mean magnetic field is log₁₀[B/G] = 12.44 and standard deviation is σ_B = 0.44. Magnetars and central compact objects do not follow the same distribution. The mean initial period is |$\log _{10} P_0 [P / \mathrm{s}] = -1.04_{-0.2}^{+0.15}$| and standard deviation is |$\sigma _{\rm p} = 0.53_{-0.08}^{+0.12}$|⁠. We show that the normal distribution does not describe the initial periods of NSs sufficiently well. Parameters of the initial period distribution are not sensitive to the exact value of the braking index.

We analyse different possibilities to explain the wide initial spin period distribution of radio pulsars presented by Noutsos et al. With a population synthesis modelling, we demonstrate that magnetic field decay can be used to interpret the difference between the recent results by Noutsos et al and those by Popov and Turolla, where a much younger population of neutron stars associated with supernova remnants with known ages has been studied. In particular, an exponential field decay with τ_mag = 5 Myr can produce a ‘tail’ in the reconstructed initial spin period distribution up to P₀ > 1 s starting with a standard Gaussian with 〈P₀〉 = 0.3 s and σ_P = 0.15 s. Another option to explain the difference between initial spin period distributions from Noutsos et al. and Popov and Turolla – the emerging magnetic field – is also briefly discussed.