This paper introduces the concept of isogeometric analysis, a method that extends finite element analysis by using functions from computer-aided design (CAD) systems to represent complex engineering geometries exactly. The authors review the fundamentals of isogeometric analysis and apply it to various structural models, including rods, thin beams, membranes, and thin plates. They explore the $k$-refinement technique, which produces more accurate and robust results compared to finite elements. The paper also discusses the elimination of "optical" branches in frequency spectra through nonlinear parameterization, which helps avoid Gibbs phenomena in wave propagation problems. The NASA Aluminum Testbed Cylinder is used as a geometrically exact model, and the computed frequencies and mode shapes are compared favorably with experimental results. The authors conclude by highlighting the advantages of isogeometric analysis in structural vibration analysis, particularly in terms of accuracy and efficiency.This paper introduces the concept of isogeometric analysis, a method that extends finite element analysis by using functions from computer-aided design (CAD) systems to represent complex engineering geometries exactly. The authors review the fundamentals of isogeometric analysis and apply it to various structural models, including rods, thin beams, membranes, and thin plates. They explore the $k$-refinement technique, which produces more accurate and robust results compared to finite elements. The paper also discusses the elimination of "optical" branches in frequency spectra through nonlinear parameterization, which helps avoid Gibbs phenomena in wave propagation problems. The NASA Aluminum Testbed Cylinder is used as a geometrically exact model, and the computed frequencies and mode shapes are compared favorably with experimental results. The authors conclude by highlighting the advantages of isogeometric analysis in structural vibration analysis, particularly in terms of accuracy and efficiency.