Gamma-ray bursts (GRBs) were first detected in 1960s. Since then, GRBs have been observed extensively in gamma-rays. The afterglows of several hundred GRBs were also detected, providing us with abundant clues for understanding the physics of GRBs. However, X-ray observations of GRBs at the prompt burst phase are still quite lacking. These very early X-ray observations should be very important for probing the activities of GRB central engines. The Einstein Probe, a unique satellite that works mainly in soft X-rays (0.5–4 keV) with a large field of view (1.1 sr), will open up a new window for us to observe the prompt X-ray emission of GRBs. In this article, we review the current status of the studies on some special kinds of GRBs such as those rich in X-rays and those baring special emission components. Especially, we investigate possible contributions of EP to the studies of X-ray flashes (XRFs), low-luminosity GRBs, ultra-long GRBs, and the precursors of GRBs. It is found that: (1) EP is expected to detect about 810 GRB-like events, among which 95% are soft XRFs and only 1% are typical GRBs; (2) EP could detect 0.2–8 low-luminosity GRBs each year (the exact detection rate is strongly dependent on the exact spectral distribution of low luminosity GRBs, which is still largely uncertain currently); (3) EP would detect at least 20–200 ultra-long GRBs each year; (4) EP may be able to detect a lot of GRB precursors (at least 80 events per year), which are still poorly understood. We believe that the EP satellite will lead to encouraging discovers and will deepen our understanding of GRBs in various aspects such as their classifications, their progenitors, the triggering mechanisms and the beaming features, etc. We also note that the EP satellite has great advantages on its wide field of view and good sensitivity, over other similar world-wide instruments. We suggested that the project should be carried out as soon as possible.