The explosive death of a star as a supernova is one of the most dramatic events in the Universe. Supernovae have an outsized impact on many areas of astrophysics: they are major contributors to the chemical enrichment of the cosmos and significantly influence the formation of subsequent generations of stars and the evolution of galaxies. Here we review the observational properties of thermonuclear supernovae, exploding white dwarf stars resulting from the stellar evolution of low-mass stars in close binary systems. The best known objects in this class are type Ia supernovae (SN Ia), astrophysically important in their application as standardisable candles to measure cosmological distances and the primary source of iron group elements in the Universe. Surprisingly, given their prominent role, SN Ia progenitor systems and explosion mechanisms are not fully understood; the observations we describe here provide constraints on models, not always in consistent ways. Recent advances in supernova discovery and follow-up have shown that the class of thermonuclear supernovae includes more than just SN Ia, and we characterise that diversity in this review.The modern classification scheme for supernovae traces back to Minkowski 1 who in 1941 split "Type I" from "Type II" supernovae based on optical spectra. Further subdivision of these basic classes has continued on an empirical basis 2, 3 , and in our review we describe the observational properties of what are now called SN Ia, along with other similar objects. The observational classification effort arises from a desire for physical understanding of these objects, explaining our use of the term thermonuclear supernovae in the title. That categorisation is based on the explosion mechanism: objects where the energy released in the explosion is primarily the result of thermonuclear fusion. Given our current state of knowledge, we could equally well call this a review of the observational properties of white dwarf supernovae, a categorisation based on the kind of object that explodes. This is contrasted with core-collapse or massive star supernovae, respectively, in the explosion mechanism or exploding object categorisations. Unlike those objects, where clear observational evidence exists for massive star progenitors and corecollapse (from both neutrino emission and remnant pulsars), the direct evidence for thermonuclear supernova explosions of white dwarfs is limited 4, 5 and not necessarily simply interpretable 6, 7 . Nevertheless, the indirect evidence is strong, though many open questions about the progenitor systems and explosion mechanisms remain.SN Ia are important both to the evolution of the Universe and to our understanding of it. As standardisable candles whose distance can be observationally inferred 8 , SN Ia have a starring role in the discovery of the accelerating expansion of the Universe 9, 10 and in measurement of its current expansion rate 11 . SN Ia are also major contributors to the chemical enrichment of the Universe, producing most of its iron 12 and elem...