I. The Two Basic Blowup Mechanisms -- A. The ODE mechanism -- B. The geometric blowup mechanism -- C. Combinations of the two mechanisms -- Notes -- II. First Concepts on Global Cauchy Problems -- 1. Short time existence -- 2. Lifespan and blowup criterion -- 3. Blowup or not? Functional methods -- 4. Blowup or not? Comparison and averaging methods -- Notes -- III. Semilinear Wave Equations -- 1. Semilinear blowup criteria -- 2. Maximal influence domain -- 3. Maximal influence domains for weak solutions -- 4. Blowup rates at the boundary of the maximal influence domain -- 5. An example of a sharp estimate of the lifespan -- Notes -- IV. Quasilinear Systems in One Space Dimension -- 1. The scalar case -- 2. Riemann invariants, simple waves, and L1-boundedness -- 3. The case of 2 ร{151} 2 systems -- 4. General systems with small data -- 5. Rotationally invariant wave equations -- Notes -- V. Nonlinear Geometrical Optics and Applications -- 1. Quasilinear systems in one space dimension -- 2. Quasilinear wave equations -- 3. Further results on the wave equation -- Notes