  Directional amplification, in which signals are selectively amplified
depending on their propagation direction, has attracted much attention as key
resource for applications, including quantum information processing. Recently,
several, physically very different, directional amplifiers have been proposed
and realized in the lab. In this work, we present a unifying framework based on
topology to understand non-reciprocity and directional amplification in
driven-dissipative cavity arrays. Specifically, we unveil a one-to-one
correspondence between a non-zero topological invariant defined on the spectrum
of the dynamic matrix and regimes of directional amplification, in which the
end-to-end gain grows exponentially with the number of cavities. We compute
analytically the scattering matrix, the gain and reverse gain, showing their
explicit dependence on the value of the topological invariant. Parameter
regimes achieving directional amplification can be elegantly obtained from a
topological `phase diagram', which provides a guiding principle for the design
of both phase-preserving and phase-sensitive multimode directional amplifiers.
