Fibring is a powerful mechanism for combining logics, and an essential tool for designing and understanding complex logical systems. Given two logic systems L1 and L2, their fibring L1 • L2 is the smallest logical system for the combined language which contains both L1 and L2. From the point of view of Hilbert systems, this corresponds precisely to putting together the Hilbert systems of the two logics, allowing instantiations with formulas in the joint signature along the derivations. In this presentation we give a full characterization of the consequence relation that emerges from fibring in the particular case where the logics being combined do not share any connectives. We show the power of this tool by presenting various applications on important problems regarding combined logics: (1) conservativity, (2) decidability and complexity, and (3) the preservation of many-valuedness. Joint work with C. Caleiro.