This article is dedicated to geometric structure of the Lorentz and Marcinkiewicz spaces in case of the pure atomic measure. We study complete criteria for order continuity, the Fatou property, strict monotonicity, and strict convexity in the sequence Lorentz spaces γp,w. Next, we present a full characterization of extreme points of the unit ball in the sequence Lorentz space γ1,w. We also establish a complete description up to isometry of the dual and predual spaces of the sequence Lorentz spaces γ1,w written in terms of the Marcinkiewicz spaces. Finally, we show a fundamental application of geometric structure of γ1,w to one-complemented subspaces of γ1,w.