In this paper we survey several results on the study of spacelike surfaces of constant Gaussian curvature K < 0 in Lorentz-Minkowski space of dimension (2+1). Moreover, we show that the space of entire K-surfaces with bounded second fundamental form, up to translations, is naturally parameterized by an infinite-dimensional vector space, namely the tangent space at the trivial point of universal Teichmüller space.