We study the crucial role played by the solid-state environment in determining the photon emission characteristics of a driven quantum dot. For resonant driving, we predict a phonon-enhancement of the coherently emitted radiation field with increasing driving strength, in stark contrast to the conventional expectation of a rapidly decreasing fraction of coherent emission with stronger driving. This surprising behaviour results from thermalisation of the dot with respect to the phonon bath, and leads to a nonstandard regime of resonance fluorescence in which significant coherent scattering and the Mollow triplet coexist. Off-resonance, we show that despite the phonon influence, narrowing of dot spectral sideband widths can occur in certain regimes, consistent with an experimental trend.PACS numbers: 78.67. Hc, As described by Mollow, the spectrum of light scattered from a resonantly driven two-level system (TLS) depends crucially on the relative size of the laser driving strength to the TLS radiative decay rate [1]. For weak driving, the light is predominately coherently (or elastically) scattered, resulting in a single (delta function) peak in the emission spectrum at the laser frequency. At larger driving strengths, however, coherent scattering is strongly suppressed, and the emission becomes dominated by incoherent (inelastic) scattering from the TLSlaser dressed states [2]. This results in a triple-peak structure in the spectrum, known as the Mollow triplet.While these fundamental predictions have long been confirmed in the traditional quantum optical setting of driven atoms [3], interest has turned more recently to their observation in solid-state TLSs (artificial atoms) such as semiconductor quantum dots (QDs) [4][5][6][7][8][9][10], single molecules [11], and superconducting circuits [12]. In the particular case of QDs, many of the archetypal features of atomic quantum optics have now been demonstrated, such as resonance fluorescence [4][5][6][7][8][9][10], coherent population oscillations [7,[13][14][15], photon anti-bunching [16,17], and two-photon interference [18][19][20]. Aside from being of fundamental interest, these observations also pave the way towards using QDs as efficient single photon sources [21][22][23][24], and for other quantum technologies [25].Thus, under appropriate conditions, the emission properties of a driven QD can bear close resemblance to the more idealised case of a driven atom in free space. QDs are, nevertheless, unavoidably coupled to their surrounding solid-state environments. For coherentlydriven (ground state) excitonic transitions in typical arsenide QDs, coupling to acoustic phonons has been demonstrated to dominate the QD-environment interaction [14,15], leading to the appearance of an excitationinduced dephasing contribution with a rate that varies with the square of the Rabi frequency (dot-laser coupling strength) [9,14,15,26]. This driving dependence is theoretically understood as resulting from phonons that induce transitions between the dressed states of the QD at...