Less than 1 billion year after the Big Bang (< 8% of today’s age of the Universe), the cosmic star formation rate and stellar mass density of galaxies increased by more than one order of magnitude, a gradient steeper than at any other time. Hence it is expected that galaxies during this early rapid growth phase show significantly different spatial and physical properties compared to galaxies at later times. I present how observations with Spitzer, the Atacama Large Millimeter Array (ALMA), and the HST help us to study and understand this important corner-stone phase of galaxy evolution. With forward modeling of the Spitzer broad-band colors, we measure the fluxes of optical emission lines in galaxies out to z = 6. This allows us to witness the rapid growth of these infant galaxies, to study (on a basic level) their inter-stellar medium (ISM) conditions, and to quantify their contribution to the reionization of hydrogen at z > 6. Observation with ALMA of the C+ emission at 158μm and the continuum bracketing it enables a deep insight into the dust and far-infrared properties of these galaxies. Our very recent observations of a dozen galaxies at z = 6 reveal a population that has a large range in evolutionary stages already 1 billion years after the Big Bang. With our 70h ALMA program to extend these measurements to >100 galaxies at z = 4-6 (ALPINE) we will study this population in more detail.