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Despite its power as the conceptual basis for a huge range of physical phenomena in atomic and subatomic physics, quantum mechanics still suffers from a lack of clarity regarding the physical meaning of its fundamental theoretical concepts, such as those of quantum state and of quantum theoretical quantities or variables, dealt with by the known mathematical-theoretical rules. These concepts have generally been considered as not giving a direct description of physical systems, for they do not correspond to what is usually understood by 'physical states' or 'physical properties', notably characterized by definite numerical values such as those obtained from measurement. The situation has been tentatively expressed in terms of various 'interpretations', conceived not only with regard to the physical meanings of mathematical quantities, but also in connection with philosophical statements about 'physical reality'. The question of whether or not quantum theoretical quantities describe definite physical systems existing in nature has therefore generally been considered as beyond the possibilities of physics, because of the definitions that are commonly taken for 'physical state' and 'physical quantity'. We sketch the main ideas on this problem and propose a possible way out of the puzzle, in terms of an extension of the meaning given to the concepts of the physical state and physical quantity of a system, which would allow one, without any theoretical change in quantum mechanics, to speak consistently of real quantum systems as having definite physical properties.