This book addresses the idealised problem posed by homogeneous, isotropic turbulence. It is written from the perspective of a theoretical physicist, but is designed to be accessible to all researchers in turbulence, both theoretical and experimental, and from all disciplines. At the end of reading this book I felt exuberant in several distinct ways: progress has been made, but there are still vast areas of turbulence that are hardly understood; there is beautiful mathematics underpinning the analyses in HIT in particular and turbulence in general; and one has a sense of optimism that theoretical progress is gaining momentum.