Four-body distorted wave theory appropriate for nucleon-nucleus reactions leading to 3-body continuum excitations of two-neutron Borromean halo nuclei is developed. The peculiarities of the halo bound state and 3-body continuum are fully taken into account by using the method of hyperspherical harmonics. The procedure is applied for A = 6 test-bench nuclei; thus we report detailed studies of inclusive cross sections for inelastic 6 He(p,p ′ ) 6 He * and charge-exchange 6 Li(n,p) 6 He * reactions at nucleon energy 50 MeV. The theoretical low-energy spectra exhibit two resonance-like structures. The first (narrow) is the excitation of the well-known 2 + three-body resonance. The second (broad) bump is a composition of overlapping soft modes of multipolarities 1 − , 2 + , 1 + , 0 + whose relative weights depend on transferred momentum and reaction type. Inelastic scattering is the most selective tool for studying the soft dipole excitation mode.Recent success in developing experimental methods for dripline nuclei, that in particular allow exploration of halo phenomena in light nuclei, has put on the agenda a need for appropriate theoretical methods which take into account the peculiarities of weakly bound and spatially extended systems. For Borromean two-neutron halo nuclei ( 6 He, 11 Li, etc.) an understanding of the essential halo structure has been obtained in the framework of 3-body models [1]. Reactions involving these nuclei present however, at least a 4-body problem. The direct solution of 4-body systems is extremely difficult, and approximate methods are required. For high energy elastic scattering and relativistic fragmentation of Borromean halo nuclei, a 4-body Glauber method has been developed [2,3]. For Coulomb breakup or electromagnetic dissociation (EMD) the first order (Alder-Winther) perturbation theory or an equivalent semiclassical treatment [4] has been used, but with exact 3-body continuum wave functions [5][6][7]. Also for the (anti)neutrino induced reactions on 6 Li populating the 6 He and 6 Be 3-body continua, have proper final state wave functions recently been used [8].The most reliable information on properties of halo nuclei, especially for the low-lying part of excitation spectra, is experimentally obtainable by intermediate energy elastic and inelastic scattering and charge-exchange reactions. The distorted wave theory is the most common way to analyse such processes [9], but for halo systems their spatial granularity as well as peculiarities of their quantum structure have to be taken into account. The 3-body interaction dynamics defines the low-lying part of excitation spectra, in particular the soft modes of Borromean systems, and has to be treated properly. Until now, only the hyperspherical harmonics (HH) method [10] is able to provide a formulation of the scattering theory to the 3-body Borromean continuum. The Faddeev equations technique [11] has been developed to investigate breakup of 3-nucleon systems, but has hitherto not been applied to investigate the continuum in Borromean...