Gilles Chabrier reviews recent determinations of the present-day mass function (PDMF) and initial mass functions (IMF) in various Galactic components, including the disk, spheroid, young and globular clusters, and early star formation conditions. The IMF is found to depend weakly on the environment and is well described by a power-law form for masses above 1 solar mass (M☉) and a lognormal form below, except for early star formation. The disk IMF for single objects has a characteristic mass around 0.08 M☉ and a logarithmic mass variance of ~0.7, while the IMF for multiple systems has a characteristic mass of ~0.2 M☉ and a variance of ~0.6. The extension of the single IMF into the brown dwarf regime is consistent with current estimates of L- and T-dwarf densities, yielding a disk brown dwarf number density comparable to the stellar one (~0.1 pc⁻³). The IMF of young clusters is consistent with the disk field IMF, confirming that young star clusters and disk field stars represent the same stellar population. Dynamical effects, leading to the depletion of the lowest-mass objects, become significant for ages >130 Myr. The spheroid IMF is less well understood, with large metallicity spreads in photometric samples remaining puzzling. Recent observations suggest a continuous kinematic shear between the thick-disk and spheroid populations, allowing only an upper limit for the spheroid mass density and IMF. The spheroid IMF is similar to that of globular clusters and is well represented by a lognormal form with a characteristic mass slightly larger than the disk (~0.2-0.3 M☉), excluding a significant brown dwarf population. The IMF for early star formation remains undetermined, but observational constraints suggest it does not extend below ~1 M☉. These results suggest a characteristic mass for star formation that decreases with time, from conditions at large redshift to those of the spheroid or thick-disk, to present-day conditions. These conclusions remain speculative due to large uncertainties in spheroid and early star IMF determinations. The IMFs allow a reasonably robust determination of the Galactic present-day and initial stellar and brown dwarf contents, with important galactic implications for mass-to-light ratio determinations. The mass-to-light ratios from the disk and spheroid IMFs are smaller than the Salpeter IMF, agreeing with recent dynamical determinations. Theoretical models based on a Jeans-type mechanism fail to meet observational constraints, while recent simulations of compressible turbulence reproduce both qualitatively and quantitatively the stellar and substellar IMF, providing an appealing theoretical foundation. Star formation is induced by the dissipation of large-scale turbulence to smaller scales through radiative MHD shocks, producing filamentary structures. These shocks create local, non-equilibrium structures with large density contrasts, which collapse under the combined influence of turbulenceGilles Chabrier reviews recent determinations of the present-day mass function (PDMF) and initial mass functions (IMF) in various Galactic components, including the disk, spheroid, young and globular clusters, and early star formation conditions. The IMF is found to depend weakly on the environment and is well described by a power-law form for masses above 1 solar mass (M☉) and a lognormal form below, except for early star formation. The disk IMF for single objects has a characteristic mass around 0.08 M☉ and a logarithmic mass variance of ~0.7, while the IMF for multiple systems has a characteristic mass of ~0.2 M☉ and a variance of ~0.6. The extension of the single IMF into the brown dwarf regime is consistent with current estimates of L- and T-dwarf densities, yielding a disk brown dwarf number density comparable to the stellar one (~0.1 pc⁻³). The IMF of young clusters is consistent with the disk field IMF, confirming that young star clusters and disk field stars represent the same stellar population. Dynamical effects, leading to the depletion of the lowest-mass objects, become significant for ages >130 Myr. The spheroid IMF is less well understood, with large metallicity spreads in photometric samples remaining puzzling. Recent observations suggest a continuous kinematic shear between the thick-disk and spheroid populations, allowing only an upper limit for the spheroid mass density and IMF. The spheroid IMF is similar to that of globular clusters and is well represented by a lognormal form with a characteristic mass slightly larger than the disk (~0.2-0.3 M☉), excluding a significant brown dwarf population. The IMF for early star formation remains undetermined, but observational constraints suggest it does not extend below ~1 M☉. These results suggest a characteristic mass for star formation that decreases with time, from conditions at large redshift to those of the spheroid or thick-disk, to present-day conditions. These conclusions remain speculative due to large uncertainties in spheroid and early star IMF determinations. The IMFs allow a reasonably robust determination of the Galactic present-day and initial stellar and brown dwarf contents, with important galactic implications for mass-to-light ratio determinations. The mass-to-light ratios from the disk and spheroid IMFs are smaller than the Salpeter IMF, agreeing with recent dynamical determinations. Theoretical models based on a Jeans-type mechanism fail to meet observational constraints, while recent simulations of compressible turbulence reproduce both qualitatively and quantitatively the stellar and substellar IMF, providing an appealing theoretical foundation. Star formation is induced by the dissipation of large-scale turbulence to smaller scales through radiative MHD shocks, producing filamentary structures. These shocks create local, non-equilibrium structures with large density contrasts, which collapse under the combined influence of turbulence