Wireless communications with extremely large-scale array (XL-array) correspond to systems whose antenna sizes are so large that conventional modelling assumptions, such as uniform plane wave (UPW) impingement, are longer valid. This paper studies the mathematical modelling and performance analysis of XL-array communications. By deviating from the conventional modelling approach that treats the array elements as sizeless points, we explicitly model their physical area/aperture, which enables a unified modelling for the classical discrete antenna arrays and the emerging continuous surfaces. As such, a generic array/surface model that accurately takes into account the variations of signal phase, amplitude and projected aperture across array elements is proposed. Based on the proposed model, a closed-form expression of the resulting SNR with the optimal single-user MRC/MRT beamforming is derived. The expression reveals that instead of scaling linearly with the antenna number M as in conventional UPW modelling, the SNR with the more generic model increases with M with diminishing return, which is governed by the collective properties of the array, such as the array occupation ratio and the physical sizes of the array along each dimension, while irrespective of the properties of the individual array element. Additionally, we have derived an alternative insightful expression for the optimal SNR in terms of the vertical and horizontal angular spans. Furthermore, we also show that our derived results include the far-field UPW modelling as a special case. One important finding during the study of far-field approximation is the necessity to introduce a new distance criterion to complement the classical Rayleigh distance, termed uniform-power distance (UPD), which concerns the signal amplitude/power variations across array elements, instead of phase variations as for Rayleigh distance.