We report broadband characterization of the propagation of light through a multiply scattering medium by means of its Multi-Spectral Transmission Matrix. Using a single spatial light modulator, our approach enables the full control of both spatial and spectral properties of an ultrashort pulse transmitted through the medium. We demonstrate spatiotemporal focusing of the pulse at any arbitrary position and time with any desired spectral shape. Our approach opens new perspectives for fundamental studies of light-matter interaction in disordered media, and has potential applications in sensing, coherent control and imaging.Propagation of coherent light through a scattering medium produces a speckle pattern at the output [1], due to light scrambling by multiple scattering events [2]. The phase and amplitude information of the light are spatially mixed, thus limiting resolution, depth and contrast of most optical imaging techniques. Ultrashort pulses, generated by broadband modelocked lasers, are very useful for multiphotonic imaging and non-linear physics [3][4][5]. In the temporal domain, an ultrashort pulse is temporally broadened during propagation in a scattering medium, due to the long dwell time within it [6,7], which therefore limits its range of applications.However, this scattering process is linear and deterministic. Therefore, one can control the input wavefront to design the output field. In this respect, spatial light modulators (SLMs) offer more than a million degrees of freedom to control the propagation of coherent light. These systems have played an important role in the development of wavefront shaping techniques to manipulate light in complex media. Iterative optimization algorithm [8][9][10] and phase conjugation methods [11,12] have been proposed to focus light at a given output position, an essential ingredient for imaging. An alternative method for light control is the optical transmission matrix (TM). The TM is a linear operator that links the input field (SLM) to the output field (CCD camera) [1,13]. The measurement of the TM allows imaging through [15] or inside a scattering medium [16], and potentially access mesoscopic properties of the system [17].The possibility of shaping the pulse in time is also essential for coherent control [18]. Temporally, photons exit a scattering medium at different times, giving rise to a broadened pulse at its output [7,19]. Temporal spreading of the original pulse is characterized by a confinement time τ m [20] related to the Thouless time [21]. Equivalently, from a spectral point of view, the scattering medium responds differently for distinct spectral components of an ultrashort pulse, with a spectral correlation bandwidth ∆ω m ∝ 1/τ m , giving rise to a very complex spatio-temporal speckle pattern [22][23][24][25]. With a single SLM, one can manipulate spatial degrees of freedom to adjust the delay between different optical paths. Therefore spatial and temporal distortions can be both compensated using wavefront shaping techniques. This approach allows th...