Different aspects of specific models in supersymmetry as well as constraints on decaying dark matter are analysed in this thesis. In chapter 1 we give a general introduction to supersymmetry, and briefly discuss some of the concepts that are used throughout the thesis. In chapter 2 we present a version of Gauge Mediated Supersymmetry Breaking which preserves an $R$-symmetry---the gauginos are Dirac particles, the $A$-terms are zero, and there are four Higgs doublets. This offers an alternative way for gauginos to acquire mass in the supersymmetry-breaking models of Intriligator, Seiberg, and Shih \cite{Intriligator:2006dd} . Additionally, we investigate the possibility of using $R$-symmetric gauge mediation to realise the spectrum and large sfermion mixing of the model of Kribs, Poppitz, and Weiner \cite{Kribs:2007ac}. In chapter 3 we investigate the Higgs sector of the $R$-symmetric model presented in chapter 2. Furthermore, a scan of the parameter space and sample spectra are provided. Other attributes like the tuning of the model are discussed. In chapter 4 we present a complete analysis of the cosmological constraints on decaying dark matter. In order to do this, we have updated and extended previous analyses to include Lyman-$\alpha$ forest, large scale structure, and weak lensing observations. Astrophysical constraints are not considered in this thesis. The bounds on the lifetime of decaying dark matter are dominated by either the late-time integrated Sachs-Wolfe effect for the scenario with weak reionization, or CMB polarisation observations when there is significant reionization. For the respective scenarios, the lifetimes for decaying dark matter are $\Gamma^{-1} \gtrsim 100$ Gyr and $ (f \Gamma) ^{-1} \gtrsim 5.3 \times 10^8$ Gyr (at 95.4\% confidence level), where the phenomenological parameter $f$ is the fraction of the decay energy deposited in baryonic gas. This allows us to constrain particle physics models with dark matter candidates through investigation of dark matter decays into Standard Model particles via effective operators. For decaying dark matter of $\sim 100$ GeV mass, we found that the size of the coupling constant in the effective dimension-4 operators responsible for dark matter decay has to generically be $ \lesssim 10^{-22}$.

Different aspects of specific models in supersymmetry as well as constraints on decaying dark matter are analysed in this thesis. In chapter 1 we give a general introduction to supersymmetry, and briefly discuss some of the concepts that are used throughout the thesis. In chapter 2 we present a version of Gauge Mediated Supersymmetry Breaking which preserves an $R$-symmetry---the gauginos are Dirac particles, the $A$-terms are zero, and there are four Higgs doublets. This offers an alternative way for gauginos to acquire mass in the supersymmetry-breaking models of Intriligator, Seiberg, and Shih \cite{Intriligator:2006dd} . Additionally, we investigate the possibility of using $R$-symmetric gauge mediation to realise the spectrum and large sfermion mixing of the model of Kribs, Poppitz, and Weiner \cite{Kribs:2007ac}. In chapter 3 we investigate the Higgs sector of the $R$-symmetric model presented in chapter 2. Furthermore, a scan of the parameter space and sample spectra are provided. Other attributes like the tuning of the model are discussed. In chapter 4 we present a complete analysis of the cosmological constraints on decaying dark matter. In order to do this, we have updated and extended previous analyses to include Lyman-$\alpha$ forest, large scale structure, and weak lensing observations. Astrophysical constraints are not considered in this thesis. The bounds on the lifetime of decaying dark matter are dominated by either the late-time integrated Sachs-Wolfe effect for the scenario with weak reionization, or CMB polarisation observations when there is significant reionization. For the respective scenarios, the lifetimes for decaying dark matter are $\Gamma^{-1} \gtrsim 100$ Gyr and $ (f \Gamma) ^{-1} \gtrsim 5.3 \times 10^8$ Gyr (at 95.4\% confidence level), where the phenomenological parameter $f$ is the fraction of the decay energy deposited in baryonic gas. This allows us to constrain particle physics models with dark matter candidates through investigation of dark matter decays into Standard Model particles via effective operators. For decaying dark matter of $\sim 100$ GeV mass, we found that the size of the coupling constant in the effective dimension-4 operators responsible for dark matter decay has to generically be $ \lesssim 10^{-22}$.