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As one of Stage IV space-based telescopes, China Space Station Telescope (CSST) can perform photometric and spectroscopic surveys simultaneously to efficiently explore the Universe in extreme precision. In this work, we investigate several powerful CSST cosmological probes, including cosmic shear, galaxy-galaxy lensing, photometric and spectroscopic galaxy clustering, and number counts of galaxy clusters, and study the capability of these probes by forecasting the results of joint constraints on the cosmological parameters. By referring to real observational results, we generate mock data and estimate the measured errors based on CSST observational and instrumental designs. To study the systematical effects on the results, we also consider a number of systematics in CSST photometric and spectroscopic surveys, such as the intrinsic alignment, shear calibration uncertainties, photometric redshift uncertainties, galaxy bias, non-linear effects, instrumental effects, etc. The Fisher matrix method is used to derive the constraint results from individual or joint surveys on the cosmological and systematical parameters. We find that the joint constraints by including all these CSST cosmological probes can significantly improve the results from current observations by one order of magnitude at least, which gives $Ω_m$ and $σ_8$ $<$1% accuracy, and $w_0$ and $w_a$ $<$5% and 20% accuracies, respectively. This indicates that the CSST photometric and spectroscopic multi-probe surveys could provide powerful tools to explore the Universe and greatly improve the studies of relevant cosmological problems.