  A universal word for a finite alphabet $A$ and some integer $n\geq 1$ is a
word over $A$ such that every word in $A^n$ appears exactly once as a subword
(cyclically or linearly). It is well-known and easy to prove that universal
words exist for any $A$ and $n$. In this work we initiate the systematic study
of universal partial words. These are words that in addition to the letters
from $A$ may contain an arbitrary number of occurrences of a special `joker'
symbol $\Diamond\notin A$, which can be substituted by any symbol from $A$. For
example, $u=0\Diamond 011100$ is a linear partial word for the binary alphabet
$A=\{0,1\}$ and for $n=3$ (e.g., the first three letters of $u$ yield the
subwords $000$ and $010$). We present results on the existence and
non-existence of linear and cyclic universal partial words in different
situations (depending on the number of $\Diamond$s and their positions),
including various explicit constructions. We also provide numerous examples of
universal partial words that we found with the help of a computer.
