The spherically symmetric reduction of higher-dimensional Einstein-scalar theory leads to a lower-dimensional dilatonic gravity with a dilaton coupled scalar [for example, from a four-dimensional (4D) to a 2D system]. We calculate the trace anomaly and anomaly-induced effective action for 2D and 4D dilaton coupled scalars. The large-$N$ effective action for 2D quantum dilaton-scalar gravity is also found. These 2D results may be applied to the analysis of 4D spherical collapse. The role of new, dilaton-dependent terms in the trace anomaly for 2D black holes and Hawking radiation is investigated in some specific models of dilatonic gravity which represent a modification of the Callan-Giddings-Harvey-Strominger model. The conformal sector for 4D dilatonic gravity is constructed. The quantum back reaction of dilaton coupled matter is briefly discussed (it may lead to an inflationary universe with a nontrivial dilaton).