This book consists of two lecture notes on geometric flow equations (O. Schnürer) and Lorentzian geometry  - holonomy, spinors and Cauchy Problems (H. Baum and T. Leistner) written by leading experts in these fields.It grew out of the summer school “Geometric flows and the geometry of space-time” held in  Hamburg (2016) and provides an excellent introduction for students of mathematics and theoretical physics to important themes of current research in global analysis, differential geometry and mathematical physics