A De Bruijn cycle is a cyclic sequence in which every word of length $n$ over an alphabet $\mathcal{A}$ appears exactly once. De Bruijn tori are a two-dimensional analogue. Motivated by recent progress on universal partial cycles and words, which shorten De Bruijn cycles using a wildcard character, we introduce universal partial tori and matrices. We find them computationally and construct infinitely many of them using one-dimensional variants of universal cycles, including a new variant called a universal partial family.