  In this paper, we consider the problem of guessing a sequence subject to a
distortion constraint. Specifically, we assume the following game between Alice
and Bob: Alice has a sequence $\bx$ of length $n$. Bob wishes to guess $\bx$,
yet he is satisfied with finding any sequence $\hat{\bx}$ which is within a
given distortion $D$ from $\bx$. Thus, he successively submits queries to
Alice, until receiving an affirmative answer, stating that his guess was within
the required distortion.
  Finding guessing strategies which minimize the number of guesses (the
\emph{guesswork}), and analyzing its properties (e.g., its $\rho$--th moment)
has several applications in information security, source and channel coding.
Guessing subject to a distortion constraint is especially useful when
considering contemporary biometrically--secured systems, where the "password"
which protects the data is not a single, fixed vector but rather a \emph{ball
of feature vectors} centered at some $\bx$, and any feature vector within the
ball results in acceptance.
  We formally define the guessing problem under distortion in \emph{four
different setups}: memoryless sources, guessing through a noisy channel,
sources with memory and individual sequences. We suggest a randomized guessing
strategy which is asymptotically optimal for all setups and is \emph{five--fold
universal}, as it is independent of the source statistics, the channel, the
moment to be optimized, the distortion measure and the distortion level.
