In this thesis different numerical models based on the lattice Boltzmann equation presented and tested. Therefore the models are applied to two classical two-dimensional hydrodynamical problems. Parameter studies are performed with particular regard to changes in flow dynamics at hydrodynamical instabilities. The iLBGK model [Z. Guo et al., 2000] with a D2Q9 lattice is used to study the 2D flow past a cylinder placed between two walls. The transition from a steady flow to a vortex shedding regime is analyzed by varying the Reynolds number and the distance of the cylinder to one wall. Due to interaction of the cylinder's wake with the wall vorticity, the transition is delayed as the cylinder approaches the wall. The results are compared with the findings of Zovatto & Pedrizzetti [2001]. For the simulation of thermal flows, the multi-distribution-function (MDF) approach [Z. Guo et al., 2002a; He et al., 1998] is used. This approach uses the Boussinesq approximation to separate the liquid and the thermal components of the flow, which are solved on separate lattices. Two implementations of this approach are carried out using the LBGK and MRT models [Ginzburg, 2005; Wang et al., 2013]. These thermal models are used to study the 2D Rayleigh-Bénard problem for a fixed Prandtl number Pr = 0.71. The transition from the solely conductive to the convective regime is found to be dependent on the wavenumber k of a perturbation. The usage of lateral periodic boundary conditions restricts the possible values for k, which depend on the aspect ratio of the numerical domain. Checked against theoretical results, the critical Rayleigh numbers obtained with the MRT model are found to be more accurate than those obtained with the LBGK model.