  We study $\epsilon$-representations of discrete groups by unitary operators
on a Hilbert space. We define the notion of Ulam stability of a group which
loosely means that finite-dimensional $\epsilon$-represendations are uniformly
close to unitary representations. One of our main results is that certain
lattices in connected semi-simple Lie groups of higher rank are Ulam stable.
For infinite-dimensional $\epsilon$-representations, the similarly defined
notion of strong Ulam stability is defined and it is shown that groups with
free subgroups are not strongly Ulam stable. We also study deformation rigidity
of unitary representations and show that groups containing a free subgroup are
not deformation rigid.
