This thesis investigates properties of classical and quantum spin systems on lattices. These models have been widely studied due to their relevance to condensed matter physics.

We identify the ground states of an antiferromagnetic RP2 model, these ground states are very diļæ½erent from the ferromagnetic model and there was some disagreement over their structure, we settle this disagreement.

Correlation inequalities are proved for the spin- 1/2 XY model and the ground state of the spin-1 XY model. This provides fresh results in a topic that had been stagnant and allows the proof of some new results, for example existence of some correlation functions in the thermodynamic limit.

The occurrence of nematic order at low temperature in a quantum nematic model is proved using the method of reflection positivity and infrared bounds. Previous results on this nematic order were achieved indirectly via a probabilistic representation. This result is maintained in the presence of a small antiferromagnetic interaction, this case was not previously covered.

Probabilistic representations for quantum spin systems are introduced and some consequences are presented. In particular, NĀ“eel order is proved in a bilinear-biquadratic spin-1 system at low temperature. This result extends the famous result of Dyson, Lieb and Simon [35].

Dilute spin systems are introduced and the occurrence of a phase transition at low temperature characterised by preferential occupation of the even or odd sublattice of a cubic box is proved. This result is the first of its type for such a mixed classical and quantum system. A probabilistic representation of the spin-1 Bose-Hubbard model is also presented and some consequences are proved.