Function spaces with dominating mixed smoothness We study several techniques which are well known in the case of Besov and TriebelLizorkin spaces and extend them to spaces with dominating mixed smoothness. We use the ideas of Triebel to prove three important decomposition theorems. We deal withsocalled atomic subatomic and wavelet decompositions. All these theorems have much in common. fRoughly speaking they say that a function belongs to some function space if and only if it can be decomposed into the sum of products of coefficients and corresponding building blocks where the coefficients belong to an appropriate sequence space. These decomposition theorems estabilisha veryusefulconnection between function and sequence spaces. We use them in the study of the decay of entropy numbers of compact embeddings between two function spaces of dominating mixed smoothness reducingthis problem to the same question on the sequence space level. The considered scales cover many important specific spaces (Sobolev Zygmund Besov) and we get generalisations of respective assertions of Belinsky Dinh Dung and Temlyakov.