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In this article generalized attenuated ray transforms (ART) and integralangular moments are investigated. Starting from the Radon transform, theattenuated ray transform and the longitudinal ray transform, we derive theconcept of ART-operators of order $k$ over functions defined on the phase spaceand depending on time. The ART-operators are generalized for complex-valuedabsorption coefficient as well as weight functions of polynomial andexponential type. Connections between ART operators of various orders areestablished by means of the application of the linear part of a transportequation. These connections lead to inhomogeneous differential equations oforder $(k+1)$ for the ART of order $k$. Uniqueness theorems for thecorresponding boundary-value and initial boundary-value problems are proved.Properties of integral angular moments of order $p$ are considered andconnections between the moments of different orders are deduced. A closeconnection of the considered operators with mathematical models for tomography,physical optics and integral geometry allows to treat the inversion of ART oforder $k$ as an inverse problem of determining the right-hand side of acorresponding differential equation.

Generalized attenuated ray transforms and their integral angular moments