Complex-variable theory provides insight into the nature of physical theories. For example, it provides a simple and beautiful picture of quantization and it explains the underlying reason for the divergence of perturbation theory. By using complex-variable methods one can generalize conventional Hermitian quantum theories into the complex domain. The result is an exciting new class of parity-time-symmetric (PT-symmetric) theories whose remarkable physical properties are currently under intense study by theorists and experimentalists. Many theoretical predictions have been verified in recent laboratory experiments.