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Although general relativity passes all precision tests to date, there are several reasons to go beyond the current model of gravitation and search for new fundamental physics. This means looking for new, so far undetected, fields. Scalars are the easiest fields to consider and they are ubiquitous in both extensions of the Standard Model and in alternative theories of gravity. That is why a lot of attention has been drawn towards the investigation of scalar-tensor theories, where gravity is described by both the metric tensor and a scalar field. A particularly interesting phenomenon that has recently gained increasing interest is spontaneous scalarization. In gravity theories that exhibit this mechanism, astrophysical objects are identical to their general relativistic counterpart until they reach a specific threshold, usually either in compactness, curvature or, as recently shown, in spin. Beyond this threshold, they acquire a nontrivial scalar configuration, which also affects their structure. In this thesis, we focus on the study of this mechanism in generalized scalar-tensor theories. We identify a minimal action that contains all of the terms that can potentially trigger spontaneous scalarization. We first focus on the onset of scalarization in this specific theory and determine the relevant thresholds in terms of the contributing coupling constants and the properties of the compact object. Finally, we study the effect of this model on the properties of both scalarized black holes and neutron stars, such as affecting their domain of existence or the amount of scalar charge they carry.