We define tephras and cryptotephras and their components (mainly ash-sized particles of glass ± crystals in distal deposits) and summarize the basis of tephrochronology as a chronostratigraphic correlational and dating tool for palaeoenvironmental, geological, and archaeological research. We then document and appraise recent advances in analytical methods used to determine the major, minor, and trace elements of individual glass shards from tephra or cryptotephra deposits to aid their correlation and application. Protocols developed recently for the electron probe microanalysis of major elements in individual glass shards help to improve data quality and standardize reporting procedures. A narrow electron beam (diameter ~35 μm) can now be used to analyze smaller glass shards than previously attainable. Reliable analyses of 'microshards' (defined here as glass shards <32 µm in diameter) using narrow beams are useful for fine-grained samples from distal or ultra-distal geographic locations, and for vesicular or microlite-rich glass shards or small melt inclusions. Caveats apply, however, in the microprobe analysis of very small microshards (~5 µm in diameter), where particle geometry becomes important, and of microlite-rich glass shards where the potential problem of secondary fluorescence across phase boundaries needs to be recognised. Trace element analyses of individual glass shards using laser ablation inductively coupled plasma-mass spectrometry (LA-ICP-MS), with crater diameters of 20 μm and 10 μm, are now effectively routine, giving detection limits well Highlights Advances in the microanalysis of major, minor, and trace elements of glass shards are reviewed We evaluate numerical and statistical methods for tephra correlation via glass/crystal analyses We focus on (1) differences in mean composition of samples or their range; and (2) sample variance and degree of compositional similarity to establish equivalence or not We illustrate various statistical methods and data transformations using case studies Wherever possible, such analytical data are very markedly supported and more readily interpreted by the attainment of numerical ages on tephras (Turner et al., 2011b; Green et al., 2014; Damaschke et al., 2017a). Dating techniques applied to tephras include: (i) radiometric, for example radiocarbon (14 C), fission track, luminescence, 40 Ar/ 39 Ar, U-Th-disequilibrium/U-Pb and (U-Th)/He zircon dating (e.g. Biswas et al.,