Consider a feedback system as shown in Figure 1.

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Figure 1. Block diagram of a feedback control system

The closed loop transfer function is,

(1)   \begin{equation*} A_f(j\omega) ={X_o(j\omega) \over X_i(j\omega)}= {A(j\omega) \over 1 - A(j\omega)\beta(j\omega)} \end{equation*}

The characteristic equation is \left{ 1 - A(j\omega)\beta(j\omega) = 0\right}.
For sustained oscillations at a frequency (\omega_o), the roots of the characteristic equation should be at s=\pm j\omega_o.

According to Barkhausen criterion the conditions for sustained oscillations are

(2)   \begin{equation*} |A\beta|=1 \mbox{ and } \angle{A\beta} = 0 \end{equation*}

Amplitude of oscillations are controlled by |A\beta| and frequency by \angle{A\beta}.

To start oscillations

(3)   \begin{equation*} |A\beta|>1 \mbox{ and } \angle{A\beta} = 0 \end{equation*}

The AC output voltage is limited either by voltage limiting or current limiting.

The two mechanisms have very different behavior. With voltage limiting, the output voltage begins to resemble a square wave. The odd-order harmonic distortion will increase. If the circuit is intended to provide good linear amplification or good spectral purity, this scenario is to be avoided.
With current limiting, the signal amplitude can be adjusted so that it never reaches clipping. It swings above and below VDD without distorting. It is always to build the oscillator or amplifier so that it current limits.

An oscillator oscillates only at a frequency where \angle{A\beta}=0. In that case if |A\beta|>1, what happens to amplitude of oscollations in steady-state?

Oscillator is an amplifier with positive feedback. An inverting amplifier is the basic building block of an oscillator.

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