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We solve the wavelet set existence problem. That is, we characterize the full-rank lattices $$\Gamma\subset\mathbb{R}^n$$ and invertible $$n\times n$$ matrices $$A$$ for which there exists a measurable set $$W$$ such that $$\{W+\gamma:\gamma\in\Gamma\}$$ and $$\{A^j(W):j\in\mathbb{Z}\}$$ are tilings of $$\mathbb{R}^n$$. The characterization is a non-obvious generalization of the one found by Ionascu and Wang (2006), which solved the problem in the case $n=2$. As an application of our condition and a theorem of Margulis, we also strengthen a result of Dai, Larson, and the second author on the existence of wavelet sets by showing that wavelet sets exist for matrix dilations, all of whose eigenvalues $$\lambda$$ satisfy $$|\lambda|\ge 1$$. As another application, we extend the Ionascu-Wang characterization to higher dimensions for dilations whose product of two smallest eigenvalues in absolute value is $$\ge 1$$. Finally, we show the existence of wavelet sets for all dilations $$A$$ with integer entries satisfying $$|\det A|\ne 1$$.