In this dissertation, we analyze both many- and few-body systems under external confinement with tunable interactions. First, we develop a density-renormalization approach for describing two-component fermionic systems with short-range interactions. This renormalized zero-range interaction eliminates the instabilities produced by a bare Fermi pseudopotential and provides a simple description of the interactions from the weakly interacting BCS region up to unitarity.

In the second part of the thesis, we focus on few-body systems in the BCS-BEC crossover. To obtain the solutions, we implement two different numerical techniques: a correlated-Gaussian-basis-set expansion and a fixed-node diffusion Monte Carlo technique. We also develop an innovative numerical technique for obtaining solutions to the four-body problem in the hyperspherical representation.

Our solutions provide an accurate description of few-body trapped systems. The analysis of two-, three-, and four-body systems, for instance, provides a few-body perspective on the BCS-BEC crossover problem. The analysis of the spectrum of such systems allows us to visualize important pathways for molecule formation. We then use the four-body solutions to extract key properties of the system such as the dimer-dimer scattering length and the effective range.

We also explore the qualitative change of behavior in the BCS-BEC crossover by analyzing the spectrum and structural properties. We investigate the dynamics of these few-body systems and analyze them using a Landau-Zener model. At unitarity, we study the universal properties of few-body systems and verify the absence of many-body bound states up to N=6.

Finally, we present preliminary results on the four-boson system. We analyze the structure of the spectrum and find a family of four-body states attached to the three-body thresholds. These four-body states follow the universal scaling properties of the Efimov states. We explore the collisional implications of these four-body states and find relations between the atom-dimer and dimer-dimer collisional properties. In particular, we predict that these four-body states will produce resonances in the dimer-dimer scattering length.