This paper is a study on a new kind modulation spaces M(P, Q)(R-d) and A(P, Q, r)(R-d) for indices in the range 1 < P < infinity, 1 <= Q < infinity and 1 <= r < infinity, modelled on Lorentz mixed norm spaces instead of mixed norm L-P spaces as the spaces M-m(p,q) (R-d) (Feichtinger in Modulation spaces on locally compact Abelian groups, 1983; Grochenig in Foundations of Time-Frequency Analysis. Birkh auser, Boston, 2001), and Lorentz spaces as the spaces M(p, q)(R-d) (Gurkanhin J Math Kyoto Univ 46:595-616, 2006). First, we prove the main properties of these spaces. Later, we describe the dual spaces and determine the multiplier spaces for both of them. Moreover, we investigate the boundedness of Weyl operators and localization operators on M(P, Q)(R-d). Finally, we give an interpolation theorem for M(P, Q)(R-d).