  In the Shortest Superstring problem we are given a set of strings $S=\{s_1,
\ldots, s_n\}$ and integer $\ell$ and the question is to decide whether there
is a superstring $s$ of length at most $\ell$ containing all strings of $S$ as
substrings. We obtain several parameterized algorithms and complexity results
for this problem.
  In particular, we give an algorithm which in time $2^{O(k)}
\operatorname{poly}(n)$ finds a superstring of length at most $\ell$ containing
at least $k$ strings of $S$. We complement this by the lower bound showing that
such a parameterization does not admit a polynomial kernel up to some
complexity assumption. We also obtain several results about "below guaranteed
values" parameterization of the problem. We show that parameterization by
compression admits a polynomial kernel while parameterization "below matching"
is hard.
