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We calculate the proton and neutron spin contributions for nuclei usingsemi-empirical methods, as well as a novel hybrid \emph{abinitio}/semi-empirical method, for interpretation of experimental data. Wedemonstrate that core-polarisation corrections to \emph{ab initio} nuclearshell model calculations generally reduce discrepancies in proton and neutronspin expectation values from different calculations. We derive constraints onthe spin-dependent P,T-violating interaction of a bound proton with nucleons,which for certain ranges of exchanged pseudoscalar boson masses improve on themost stringent laboratory limits by several orders of magnitude. We derive alimit on the CPT and Lorentz-invariance-violating parameter$|\tilde{b}_{\perp}^p| < 7.6 \times 10^{-33}$ GeV, which improves on the moststringent existing limit by a factor of 8, and demonstrate sensitivities to theparameters $\tilde{d}_{\perp}^p$ and $\tilde{g}_{ D\perp}^p$ at the level $\sim10^{-29} - 10^{-28}$ GeV, which is a one order of magnitude improvementcompared to the corresponding existing sensitivities. We extend previousanalysis of nuclear anapole moment data for Cs to obtain new limits on severalother CPT and Lorentz-invariance-violating parameters: $\left|b_0^p \right| < 7\times 10^{-8}$ GeV, $\left|d_{00}^p \right| < 8 \times 10^{-8}$, $\left|b_0^n\right| < 3 \times 10^{-7}$ GeV and $\left|d_{00}^n \right| < 3 \times10^{-7}$.

Nuclear spin-dependent interactions: Searches for WIMP, Axion and Topological Defect Dark Matter, and Tests of Fundamental Symmetries