We have developed a method based on wavelet transforms (WT) to detect sources in astronomical images obtained with photon-counting detectors, such as X-ray images. The WT is a multiscale transform which is suitable for detection and analysis of interesting image features (sources) spanning a range of sizes. This property of the WT is particularly well suited to the case where the Point-Spread Function is strongly varying across the image, and it is also effective in the detection of extended sources. The method allows to measure source count rates, sizes, and ellipticity, with their errors. Care has been taken in the assessment of thresholds for detection, in the WT space, at any desired confidence level, through a detailed semi-analytical study of the statistical properties of noise in wavelet-transformed images. The method includes the use of exposure maps to handle sharp background gradients produced by a non-uniform exposure across the detector, which would otherwise yield many spurious detections. The same method is applied to evaluate upper limits to the count rate of undetected objects in the field of view, allowing a sensitivity map for each observation to be constructed.