Analysis of the Planck 2018 data set indicates that the statistical
properties of the cosmic microwave background (CMB) temperature anisotropies
are in excellent agreement with previous studies using the 2013 and 2015 data
releases. In particular, they are consistent with the Gaussian predictions of
the $\Lambda$CDM cosmological model, yet also confirm the presence of several
so-called "anomalies" on large angular scales. The novelty of the current
study, however, lies in being a first attempt at a comprehensive analysis of
the statistics of the polarization signal over all angular scales, using either
maps of the Stokes parameters, $Q$ and $U$, or the $E$-mode signal derived from
these using a new methodology (which we describe in an appendix). Although
remarkable progress has been made in reducing the systematic effects that
contaminated the 2015 polarization maps on large angular scales, it is still
the case that residual systematics (and our ability to simulate them) can limit
some tests of non-Gaussianity and isotropy. However, a detailed set of null
tests applied to the maps indicates that these issues do not dominate the
analysis on intermediate and large angular scales (i.e., $\ell \lesssim 400$).
In this regime, no unambiguous detections of cosmological non-Gaussianity, or
of anomalies corresponding to those seen in temperature, are claimed. Notably,
the stacking of CMB polarization signals centred on the positions of
temperature hot and cold spots exhibits excellent agreement with the
$\Lambda$CDM cosmological model, and also gives a clear indication of how
Planck provides state-of-the-art measurements of CMB temperature and
polarization on degree scales.