  In this thesis, I go through the well-known solutions to the one and
two-particle systems trapped in a quantum harmonic oscillator and then continue
to the three, four and many-body quantum systems. This is done by developing
new analytical models and numerical methods both for the few- and many-body
systems. One-dimensional systems are very interesting in a sense that particles
aligned on a line can only change seats by going through each other. This
property can be exploited in the strongly interacting regime, where particles
are forced to sit in a specific configuration, which can be easily manipulated.
The knowledge of how and where the particles are can be exploited in future
quantum applications. In short, the thesis is about establishing a solid
knowledge about everything that one needs to know about the one-dimensional
few- and many-component interacting quantum systems trapped in harmonic
oscillator potentials.
