Sigurdur Helgason is a leading expert in harmonic analysis and integral geometry on symmetric spaces. His work has had, and continues to have, a profound influence on the field. Helgason's work is marked by an interplay of analysis, geometry, and representation theory. The articles collected here cover invariant differential operators, geometric properties of solutions to differential equations on symmetric spaces, double fibrations in integral geometry, spherical functions and spherical transforms, duality for symmetric spaces, representation theory, and the Fourier transform on G/K. The papers are supplemented by an introductory essay by Helgason. This Selecta of Sigurdur Helgason's important papers will be a valuable resource to research mathematicians and graduate students.