Possibilistic logic is a weighted logic introduced and developed since the mid-1980s, in the setting of artificial intelligence, with a view to develop a simple and rigorous approach to automated reasoning from uncertain or prioritized incomplete information. Standard possibilistic logic expressions are classical logic formulas associated with weights, interpreted in the framework of possibility theory as lower bounds of necessity degrees. Possibilistic logic handles partial inconsistency since an inconsistency level can be computed for each possibilistic logic base. Logical formulas with a weight strictly greater than this level are immune to inconsistency and can be safely used in deductive reasoning. This paper first recalls the basic features of possibilistic logic, including information fusion operations. Then, several extensions that mainly deal with the nature and the handling of the weights attached to formulas, are suggested or surveyed: the leximin-based comparison of proofs, the use of partially ordered scales for the weights, or the management of fuzzily restricted variables. Inference principles that are more powerful than the basic possibilistic inference in case of inconsistency are also briefly considered. The interest of a companion logic, based on the notion of guaranteed possibility functions, and working in a way opposite to the one of usual logic, is also emphasized. Its joint use with standard possibilistic logic is briefly discussed. This position paper stresses the main ideas only and refers to previous published literature for technical details.