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In this paper we study a class of broken ray transforms (BRT), which can be implemented with flat and/or curved detectors. In the case of two detectors we obtain an inversion formula, which involves a second-order derivative of the data and integration along characteristics. In the case of three detectors, we obtain an inversion formula, which is purely local and involves only the first-order derivatives of the data. Hence the formula solves the interior problem. Neither the object nor the source and detectors require to be rotated in order to obtain a complete data set. We also prove a theorem, which describes the range of the BRT in the case of three detectors. Finally, the results of numerical experiments are presented.