I will present a new description of the envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold in terms of equivalence classes of analytic discs. The approach implies new results. In particular, for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary projecting into the original domain. Another corollary concerns representation of certain elements of the fundamental group of the domain by boundaries of immersed analytic discs in the envelope of holomorphy. It has an analog for the case of Stein fillings of contact manifolds.