The first four chapters of this thesis are concerned with the detailed mathematical and physical properties of special minimum uncertainty states of the electromagnetic field: the “squeezed states”. These are states with reduced quantum fluctuations in certain observables (the quadrature phase amplitudes). In chapter two it is shown that the one mode and two mode squeezed states are obtained by acting on the vacuum state with elements of the sympletic group in one and two dimensions respectively. In chapter three properties of squeezed states in a quantum optical context are investigated. The complex P representation of Drummond and Gardiner is shown to be especially useful for discussing the non-classical statistics of these states, and the complex P-representation for a squeezed state is obtained. An analysis of the effect of irreversible dynamics on squeezed states is also given. In chapter four a detailed analysis of three devices proposed to produce squeezed states; degenerate parametric oscillation, two photon absorption and dispersive optical bistability, is presented. It is shown in an exact, fully quantum treatment that the reduction of fluctuations obtainable in all these devices is limited, when operated in a steady state regime. The greatest reduction of fluctuations (a factor of two) occurs in the parametric oscillator operated at threshold. The addition of a second driving field in this device enables the direction of squeezing to be changed. The final two chapters are given over to a discussion of Quantum Non Demolition (QND) measurements. In chapter five a particular QND scheme (back action evading) is analysed. This is based on a parametric amplification interaction. The complex p representation is used to discuss the non-unitary effects of damping and state reduction in a unified way. It is shown that the parametric amplifier, despite being non back action evading, is useful for making QND measurements. Finally in chapter six, two quantum counting QND schemes, based on quantum optical four wave mixing interactions, are analysed. The effect of state reduction is included, thus enabling an analysis of a measurement sequence.