  Ambit stochastics is the name for the theory and applications of ambit fields
and ambit processes and constitutes a new research area in stochastics for
tempo-spatial phenomena. This paper gives an overview of the main findings in
ambit stochastics up to date and establishes new results on general properties
of ambit fields. Moreover, it develops the concept of tempo-spatial stochastic
volatility/intermittency within ambit fields. Various types of volatility
modulation ranging from stochastic scaling of the amplitude, to stochastic time
change and extended subordination of random measures and to probability and
L\'{e}vy mixing of volatility/intensity parameters will be developed. Important
examples for concrete model specifications within the class of ambit fields are
given.
