In the next paragraphs we will present the main goals of this doctoral thesis, and will try to guide the reader through the concepts that will be discussed in what follows. Basically, we have focused on two topics: the interplay between neutral scalars and the anomalous magnetic moment (AMM) of charged leptons, and dark matter (DM). Let us begin by summarising the full structure of the thesis, which is basically divided into four parts, namely: - Part I: Introductory concepts. In Chapter 1, we will briefly review the construction and main ingredients of the Standard Model (SM) of particle physics. Despite the fact that its predictions have been tested with a great accuracy, see e.g. the electron AMM, computed at the level of parts-per-billion (ppb), there are still several open questions that call for solutions beyond the SM (BSM). Therefore, we will motivate the need for new physics (NP) presenting the most relevant unsolved problems in the theoretical framework of the SM, including a brief review on neutrino masses in the last part of the chapter. If NP involves energy scales much larger than the current energy thresholds reached at the high-energy particle colliders such as the Large Hadron Collider (LHC), one important tool for parameterising its effects on experimental observables are effective field theories (EFTs), which will also be revisited in the last part of Chapter 1. - Part II: Anomalous Magnetic Moments of charged leptons. In this part, we will give an introduction to the AMM of charged leptons. In particular, we will start with a historical review of the events that led to the concept of spin in the first section of Chapter 2. After this, we will analyse the AMM of charged leptons, focusing on its different contributions in the SM. Namely, it gets corrections from Quantum Electrodynamics (QED), Electroweak (EW) theory and Quantum Chromodynamics (QCD) or hadronic corrections. We will also present the current values of the SM prediction and the experimental measurements of the AMM of charged leptons. Related to this, the last measurement of the fine-structure constant leads to discrepancies at the level of $1.6\,\sigma$ and $4.2\,\sigma$ for electron and muon AMM, respectively. For the latter (and also for both anomalies), we will review possible explanations that are available in the literature. After the discussion about the discrepancies, we will focus on the electron AMM in Chapter 3, and how the presence of NP, in particular light neutral scalars, can pollute the extraction of the fine-structure constant from the electron AMM. We will revisit experimental bounds that constrain the parameter space of the NP, i.e. its coupling to electrons and its mass. Moreover, in the last section of the chapter, we will analyse the parameter space of new scalars and pseudo-scalars that couple to charged leptons and are able to reproduce the observed discrepancies in both the electron and the muon AMM. - Part III: Scalars in Dark Matter scenarios. In this part, we will study the other important topic of the thesis, DM. First, we will motivate the need for DM and present its main properties in Chapter 4. Encouraged by the analysis of light neutral scalars in the framework of the AMM, in Chapter 5 we will investigate the possibilities of obtaining a suitable DM candidate from a complex scalar singlet. By doing so, we will analyse the possible discrete symmetries the complex scalar should have for yielding a good (stable) DM candidate. This will provide us with several scenarios (which will be referred to as minimal models) that will be discussed in detail. In addition, we will apply the EFT approach in the last part of the chapter, where the radial part of the complex scalar will be integrated out, yielding a pseudo-Nambu-Goldstone boson (pNGB) DM candidate. Following with the connection between neutral scalars and DM, in Chapter 6 we will analyse a Majorana DM candidate that yields the correct relic abundance thanks to its annihilations into right-handed neutrinos (sterile neutrinos). In this scenario, scalars will play the role of mediators between the dark sector and sterile neutrinos, but contrary to the case of light scalars analysed in the context of AMM, here we will consider heavy scalars, whose effects will be parameterised by means of EFTs. In particular, we will list all the possible four-fermion operators at dimension six that describe the interactions between the DM candidate and sterile neutrinos, and will discuss their UV completions. We will find several UV models in which the mediators of the interactions are scalars and vector bosons. In addition, we will provide details of the matching for the considered models in dedicated appendices. Finally, we will analyse the phenomenology of the most promising UV models. In particular, one of them will generate a Majorana mass for sterile neutrinos at one loop, which will also be discussed in a dedicated appendix. - Part IV: Final remarks and conclusions. In this part, we will summarise the interpretation of the results from the different chapters of the thesis and the conclusions that can be extracted from them. As has been mentioned in the previous paragraphs, we will give details of the relevant calculations in several appendices.