Second-order models arise naturally in the study of many types of physical systems, with common examples being electrical and structural dynamics systems. In this talk, we will discuss two tightly connected problems associated with second-order systems, namely, quadratic eigenvalue problem and reduced-order modeling. We will focus on the development of numerical techniques which exploit and preserve underlying problem structures and mathematical properties for solving quadratic eigenvalue problem and extracting reduced-order model. Case studies from structural dynamics and microelectromechanical systems (MEMS) will be discussed.