Non-intrusive speech assessment metrics have garnered significant attention in recent years, and several deep learning-based models have been developed accordingly. Although these models are more flexible than conventional speech assessment metrics, most of them are designed to estimate a specific evaluation score, whereas speech assessment generally involves multiple facets. Herein, we propose a cross-domain multiobjective speech assessment model called MOSA-Net, which can estimate multiple speech assessment metrics simultaneously. More specifically, MOSA-Net is designed to estimate the speech quality, intelligibility, and distortion assessment scores of an input test speech signal. It comprises a convolutional neural network and bidirectional long short-term memory (CRNN) architecture for representation extraction, and a multiplicative attention layer and a fully connected layer for each assessment metric. In addition, cross-domain features (spectral and timedomain features) and latent representations from self-supervised learned (SSL) models are used as inputs to combine rich acoustic information from different speech representations to obtain more accurate assessments. Experimental results show that MOSA-Net can improve the linear correlation coefficient (LCC) by 0.026 (0.990 vs 0.964 in seen noise environments) and 0.012 (0.969 vs 0.957 in unseen noise environments) in perceptual evaluation of speech quality (PESQ) prediction, compared to Quality-Net, an existing single-task model for PESQ prediction, and improve LCC by 0.021 (0.985 vs 0.964 in seen noise environments) and 0.047 (0.836 vs 0.789 in unseen noise environments) in short-time objective intelligibility (STOI) prediction, compared to STOI-Net (based on CRNN), an existing single-task model for STOI prediction. Moreover, MOSA-Net, originally trained to assess objective scores, can be used as a pre-trained model to be effectively adapted to an assessment model for predicting subjective quality and intelligibility scores with a limited amount of training data. Experimental results show that MOSA-Net can improve LCC by 0.018 (0.805 vs 0.787) in mean opinion score (MOS) prediction, compared to MOS-SSL, a strong singletask model for MOS prediction. In light of the confirmed prediction capability, we further adopt the latent representations of MOSA-Net to guide the speech enhancement (SE) process and