In this paper I advocate a version of selective epistemological realism. I begin with analyzing the conditions in which a scientific model successfully and correctly represents an identified target. I stress that the realistic import of models rests on the truth of some predicative statements. I then examine the notions of objectivity and truth in order to be able to assess the reasons to believe in the existence of some unobserved objects posited by our best scientific theories and in the truth of some assertions about the properties that these objects possess. I distinguish between the properties that are observable in principle by means of instruments which enhance our perceptive capacities (the OP properties) and the properties that are beyond any possible observation by us, namely the properties which are purely theoretical (the PT properties), such as charm and strangeness in elementary particle physics. The OP properties are identical or similar to the observed properties of ordinary perceived things, such as velocity, volume and, admittedly more controversially, charge and mass. I propose four stringent requirements for rationally believing that an unobserved object posited by a theory possesses a specific property. Firstly, this property must be an OP property. Secondly, it must be measurable. Thirdly, it must play a causal role in producing the observed data. Fourthly, distinct independent methods for measuring this property must deliver concordant results. I then show that the generality and acceptability of these four criteria is grounded on a parallelism with the reasons we adduce for (rightly) believing in the existence of ordinary observable things which we don't immediately perceive such as mice, in some circumstances. However, an agnostic attitude is to be recommended with respect to the possession of PT properties by an object posited by an—even successful—scientific theory.