In the Shortest Superstring problem we are given a set of strings S= { s₁, … , s_n} and integer ℓ and the question is to decide whether there is a superstring s of length at most ℓ 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 )}poly (n) finds a superstring of length at most ℓ containing at least k strings of S. We complement this by a 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.