## 1 Introduction  
Extra virgin olive oil is one of the most significant ingredients of the Mediterranean diet, and its consumption has been significantly increased due to its high and special nutritional and sensory characteristics [1]. Virgin olive oil (VOO) is classified to quality categories as described by official standards (e.g., Codex Alimentarius, International Olive Council, and European Union) where sensory assessment is also included [2]. The sensory quality of VOO is determined by a panel test that evaluates positive and negative descriptors, according to European Union (EU) criteria (EU regulations) [3, 4]. Olive oils possessing the fruity attribute and lacking any sensory defects can be categorized as \"extra virgin\", whilst the existence and severity of sensory defects are used to verify whether the oil could be categorized as \"virgin\" or lampante [5]. Volatile compounds in olive oil can be linked to both positive and negative sensory characteristics and have a major effect on the oil'squality and, as a result, on consumer preference. Hence, determining the VOO volatile fraction is important to ensure consumer protection, but also to avoid unfair competition, which can destabilize the market and disrupt regional/national economies. In Greece, over 30 olive varieties are mainly produced [6]. Although Greece ranks third in the world in VOO production, the high-quality potential of Greek olive oils has been scarcely examined, and such information has not been communicated to the market. Crete is one of the main olive oil-producing areas in Greece and two oil cultivars, Koroneiki and Tsounati, mainly originate from there. Koroneiki, one of the most well-known and appraised olive cultivars in the world, is the predominant olive oil cultivar in Greece, accounting for about 60% of the country's olive oil production (approximately 250,000 t) [6]. The Tsounati variety (also referred to as \"Athinolia\" or \"Mastoeidis\") is less exploited and is cultivated in specific areas of Peloponnese, mainly in south Lakonia, Argolida, as well as in western Crete [7]. The quality potential of both examined cultivars can be partially revealed from the quality awards they gain from international competitions (e.g., Athena International Olive Oil Competition). It is necessary to verify the quality standards of Greek traditional high-added-value VOO to protect the final consumer from non-authentic food products and to help Greek food producers, suppliers, and service establishments compete in national and international markets [8; 9; 10]. Gas chromatography-mass spectrometry (GC-MS) has been commonly applied for the analysis of volatile organic compounds (VOCs) in EVOO, with an emphasis given in biomarker-detection, authenticity control, and issues associated with food quality and safety [11; 12; 13]. Headspace solid-phase microextraction (HS-SPME) has a number of advantages over other extraction methods, including automation and simplicity. Due to these advantages, it is usually the method of choice, with many applications in the field [14; 15; 16]. However, for this type of analysis, it is important to identify the role of all the possible factors that can influence the extraction of volatile compounds. In the present publication, sample volume, sample stirring, extraction temperature and time, and desorption time are the factors examined to reach optimal microextraction conditions. According to our knowledge, there are only two studies that focus on the optimisation of HS-SPME parameters by utilising advanced statistical techniques such as Design of Experiments (DOE) for VOO volatile compound analysis [17; 18]. The presented approach is the first that optimises five HS-SPME factors simultaneously for EVOO analysis. The overall goal was the development of an HS-SPME/GC-MS method for the analysis of a VOO volatile profile. For this purpose, an optimisation of SPME conditions was performed using DOE and Response Surface Methodology (RSM) approaches.