###### Abstract
The maximum values of saturation magnetization (M\({}_{S}\)) for magnetite (Fe\({}_{3}\)O\({}_{4}\)) and maghemite (\(\gamma\)-Fe\({}_{2}\)O\({}_{3}\)) are 98 emu/g and 82 emu/g, respectively [10], whereas the values for saturation magnetization of hematite (\(\alpha\)-Fe\({}_{2}\)O\({}_{3}\)) are usually modest and reported to be approximately 2 to 4 emu/g [22; 23; 24]. In the case of spinel ferrite MFe\({}_{2}\)O\({}_{4}\), the maximum saturation magnetization values for CoFe\({}_{2}\)O\({}_{4}\), NiFe\({}_{2}\)O\({}_{4}\), and MgFe\({}_{2}\)O\({}_{4}\) are 94 emu/g, 56 emu/g, and 31 emu/g, respectively [10]. Regarding biocompatibility, Fe\({}_{3}\)O\({}_{4}\) and \(\gamma\)-Fe\({}_{2}\)O\({}_{3}\) have been used in various formulations approved by the USA Food and Drug Administration (FDA) as a therapy for iron deficiency and as contrast agents for magnetic resonance imaging [14; 25; 26]. Magnetite (Fe\({}_{3}\)O\({}_{4}\)) also exhibits better electronic conductivity than other magnetic oxides in the same category due to its half-metallic nature with room temperature conductivity \(\sigma=200\) (\(\Omega\cdot\)cm)\({}^{-1}\)[27; 28]. In contrast, maghemite and hematite are semiconductors with bandgaps of approximately 2.0 eV, which are certainly less conductive than half-metallic materials [15; 29]. Other spinel ferrites MFe\({}_{2}\)O\({}_{4}\) are also mostly semiconductors [30; 31]. In addition, Fe\({}_{3}\)O\({}_{4}\) possesses the first-order transition of the Verwey transition (metal-insulator transition) at approximately 115-124 K, while this property is absent in maghemite [19; 32]. Due to the Verwey transition properties, Fe\({}_{3}\)O\({}_{4}\) is potentially very useful in various physical device applications. Other advantageous properties of Fe\({}_{3}\)O\({}_{4}\) are its high electrochemical activity and high theoretical capacity, which are important for energy storage device applications [33]. In addition to  
Figure 1: Numbers of articles with the keyword “Fe\({}_{3}\)O\({}_{4}\)” published in the last 20 years. Data originated from the Web of Science. (Data from 2021 were collected until September 2021). these main properties, the natural abundance, inexpensiveness, and ecofriendliness are additional advantages that allow large-scale applications of Fe\({}_{3}\)O\({}_{4}\)-based NPs. Numerous efforts to summarize progress in the synthesis, functionalization, nanoarchitectures, and applications of Fe\({}_{3}\)O\({}_{4}\)-based NPs have been reported. Several reviews have highlighted the use of iron oxide NPs (magnetite and maghemite) in biomedical applications [34; 35; 36]. The bioinspired synthesis and green biosynthesis of magnetite NPs have been summarized by [PERSON] et al. [37] and [PERSON] et al. [38]. Although the growth mechanism of Fe\({}_{3}\)O\({}_{4}\) nanostructures and their applications were reviewed by [PERSON] and coworkers in 2011 [17], numerous advances in the field have been achieved in the last 10 years. Other reviews have focused on special physical properties or effects, such as the Verwey transition [19] and exchange bias effects [16], which provide opportunities to integrate Fe\({}_{3}\)O\({}_{4}\) NPs in electronic devices and physical instruments. Recently, [PERSON] et al. highlighted the use of Fe\({}_{3}\)O\({}_{4}\) nanostructures in pollutant gas sensor systems [39], and [PERSON] et al. reviewed synthetic methods and applications of Fe\({}_{3}\)O\({}_{4}\) in multiple fields [18]. Despite the numerous available reviews, a comprehensive review focusing on the relationship of sizes and shapes (geometries) with the magnetic properties of Fe\({}_{3}\)O\({}_{4}\) NPs, synthetic methods targeting each specific size and shape of Fe\({}_{3}\)O\({}_{4}\) NPs, and preparations of appropriate nanoparticle systems for targeted applications is still needed [40; 41; 42; 43; 44]. We envision that the size-property and geometry-property relationships are very important factors contributing to the performance of Fe\({}_{3}\)O\({}_{4}\) NPs in most applications.