Constraint satisfaction problems are a central pillar of modern computational complexity theory. This monograph provides an introduction to the rapidly growing field of Quantum Hamiltonian Complexity (QHC), which includes the study of quantum constraint satisfaction problems. Over the past decade and a half, this field has witnessed fundamental breakthroughs, ranging from the establishment of a Quantum Cook-Levin Theorem to deep insights into the structure of 1D low-temperature quantum systems via so-called area laws. Quantum Hamiltonian Complexity provides the reader with a computer...