This textbook provides an overview of the basics of ultrafast molecular spectroscopy starting from time-dependent quantum mechanical perturbation theory in Hilbert space. It emphasizes the dynamics of nuclear and electronic motion, initiated and monitored by femtosecond laser pulses, which underlie the generation of nonlinear optical signals and inform their interpretation. Topics include short-pulse electronic absorption, the molecular adiabatic approximation, transient-absorption spectroscopy, vibrational adiabaticity during conformational change, femtosecond stimulated Raman spectroscopy, multi-dimensional electronic spectroscopy and wave-packet interferometry, and two-dimensional wave-packet interferometry of electronic excitation-transfer systems. The treatment is based on time-dependent quantum mechanics as it is presented in graduate-level quantum mechanics courses. It is designed to be accessible to beginning practitioners of ultrafast spectroscopy and is meant to serve as a bridge to more advanced treatises and research publications. Numerous exercises are embedded in the text to explore and expand upon the physical ideas encountered in this important research field.