Let (X, µ) be a measure space. For p, q ∈ (0, ∞] and arbitrary subsets P, Q of (0, ∞], we introduce and characterize some intersections of Lorentz spaces, denoted by IL p,Q (X, µ), IL J,q (X, µ) and IL J,Q (X, µ).2000 Mathematics Subject Classification. Primary 43A15, Secondary 43A20.