Objectives: Breast cancer is the most commonly diagnosed type of cancer among women and a common cause of cancer-related deaths. Early diagnosis and treatment of breast cancer is critical in disease prognosis. Breast density is known to have a correlation with breast cancer. In recent years, there has been an increasing interest in the investigation of computer-aided methods for early diagnosis of breast cancer. In this study, a new fully-automated deep learning-based cascaded model was proposed for breast density assessment. In the first stage, the segmentation of adipose, fibroglandular, and pectoral muscle tissues from the digitized film mammograms of the Digital Database for Screening Mammography (DDSM) was investigated using various types of U-nets. Features extracted from the breast tissue segmentation predictions were then used to assess breast density in the second stage. Material and methods: 66 and 296 mediolateral oblique mammograms were selected from DDSM dataset for segmentation and breast density assessment systems, respectively. Different U-nets with varying number of layers and filters were implemented and the model having the highest performance was determined. U-net performance was investigated using categorical cross-entropy, Dice, Tversky, Focal Tversky, and logarithmic cosine-hyperbolic Dice loss functions. The performances of U-nets having different types of connections were investigated. The performances of U-nets having pre-trained weights from VGG16, VGG19, and ResNet50 networks in the encoding path were also investigated. Segmentation results were improved by using an image processing pipeline based on morphological operators. Segmentation performance was presented in terms of accuracy, balanced accuracy, intersection over union, and Dice's similarity coefficient (DSC) metrics. The segmentation system predictions were then used to estimate mammographic density using a machine learning pipeline by extracting features related to the fibroglandular tissue percentage. Results: Using ResNet50-U-net on the test data, average DSC scores of 82.71%, 73.39%, and 95.30% were obtained for adipose, fibroglandular, and pectoral muscle tissue segmentation, respectively. The mammogram segmentation results are 3%-12% better than the current state-of-the-art DSC in the literature when considering all of the foreground tissues concurrently. A breast density classification accuracy of 76.01% was achieved on a separate mammogram dataset, which is comparable to the recent studies in the literature. Conclusion: The proposed system can be used for automatic segmentation of mammogram into adipose, fibroglandular, and pectoral muscle tissues. The segmentation model enables the estimation of the fibroglandular-adipose tissue interface, which is recently found to be an important region for breast cancer investigations. The proposed fully-automatic breast density assessment system has a comparable performance to the ones in the literature.