This paper deals with astrophysical and technological aspects of innovative very wide-field X-ray telescopes with high sensitivity still maintaining good angular resolution. The prototypes designed, developed and tested by the authors are very promising, allowing the proposals for space projects with very wide-field Lobster-eye X-ray optics to be considered for the first time. The novel telescopes will monitor the sky with unprecedented sensitivity and angular resolution of order of 1 arcmin. They are expected to contribute essentially to study of various astrophysical objects such as AGN, SNe, Gamma-ray bursts (GRBs), X-ray flashes (XRFs), galactic binary sources, stars, CVs, X-ray novae, various transient sources, etc. For example, the Lobster optics based X-ray All Sky Monitor is capable to detect around 20 GRBs and 8 XRFs yearly and this will surely significantly contribute to the related science.