I. The Two Basic Blowup Mechanisms.- A. The ODE mechanism.- 1. Systems of ODE.- 2. Strictly hyperbolic semilinear systems in the plane.- 3. Semilinear wave equations.- B. The geometric blowup mechanism.- 1. Burgers' equation and the method of characteristics.- 2. Blowup of a quasilinear system.- 3. Blowup solutions.- 4. How to solve the blowup system.- 5. How ?u blows up.- 6. Singular solutions and explosive solutions.- C. Combinations of the two mechanisms.- 1. Which mechanism takes place first?.- 2. Simultaneous occurrence 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.- a. A functional method for Burgers' equation.- b. Semilinear wave equation.- c. The Euler system.- 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 × 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.- 1.1. Formal analysis.- 1.2. Slow time and reduced equations.- 1.3. Existence, approximation and blowup.- 2. Quasilinear wave equations.- 2.1. Formal analysis.- 2.2. Slow time and reduced equations.- 2.3. Existence, null conditions, blowup.- 3. Further results on the wave equation.- 3.1. Formal analysis near the boundary of the light cone.- 3.2. Slow time and reduced equations.- 3.3. A local blowup problem.- 3.4. Asymptotic lifespan for the two-dimensional wave equation.- Notes.

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