This paper investigates wireless communications based on a new antenna array architecture, termed modular extremely large-scale array (XL-array), where an extremely large number of antenna elements are regularly arranged on a common platform in a modular manner. Each module consists of a flexible/moderate number of antenna elements, and different modules are separated with an inter-module spacing that is typically much larger than the inter-element spacing/signal wavelength for ease of deployment. By properly modelling the variations of signal phase, amplitude and projected aperture across different array modules/elements, we develop the new channel model and analyze the signal-to-noise ratio (SNR) performance of the modular XL-array based communications. Under the practical non-uniform spherical wave (NUSW) model, the closed-form expression of the maximum achievable SNR is derived in terms of key geometric parameters, including the total planar array size, module separation distances along each dimension, as well as the user's location in the three-dimensional (3D) space. Besides, the asymptotic SNR scaling laws are revealed as the number of modules along different dimensions goes to infinity. Moreover, we show that our developed near-field modelling and performance analysis include the existing ones for the collocated XL-array, the far-field uniform plane wave (UPW) model, as well as the one-dimensional (1D) modular extremely large-scale uniform linear array (XL-ULA) as special cases. Extensive simulation results are provided to validate our obtained results.