The Hardy spaces Hp in various contexts form an important class of functional spaces. Their scopes are between complex analysis and harmonic analysis. The classical Hardy spaces Hp (C +) and Hp (D), are, respectively, associated with the upper-half plane C + and the unit disc D. The classical theory for Hp (C +) and Hp (D) is a mixture of harmonic and complex analysis that has been well understood. They have wide and deep connections and applications in both pure and applied analysis. Recently they also found significant applications in signal analysis. One intention of this thesis is to extend the theory of the classical Hardy spaces Hp (C +) to higher dimensions with the several complex variables setting. Note that an alternative approach to this goal is the real method as used, for instance, in [22], also in [17]. However, some features of the Hardy space functions are naturally and best described only with the several complex variables setting ([55]). These features, include, for instance, the Fourier spectrum characterizations of the Hardy space functions, as well as rational approximations, etc. This thesis will develop the theory of higher dimensional Hardy spaces Hp on tubes. The Hardy spaces on tubes, indeed, are the most natural stage on which both holomorphic functions of several complex variables and harmonic functions come into play. In some literature, including [55] and several later ones, Hardy spaces on tubes have been studied. But the theory is far from complete. In the existing theory, apart from the very basic results such as existence of non-tangential boundary limits for Hp functions, 1 ≤ p ≤ ∞, etc., the others are mostly restricted to the H2 case, such as the Paley-Wiener theorem, and the Cauchy and Poisson integral representations, etc. Other intentions of the thesis include Hardy spaces on the unit disc. Precisely, This thesis is mainly to get new achievements in the following three aspects. 1. We give Fourier spectrum characterizations of functions in the Hardy Hp spaces on tubes for 1 ≤ p ≤ ∞ : For F ∈ L p (R n ), we show that F is the non-tangential boundary limit of a function in such a Hardy space, Hp (TΓ), where Γ is an open cone of R n and TΓ is the relevant tube in C n , if and only if the classical or distributional Fourier transform of F is supported in Γ ∗ , where Γ ∗ is the dual cone of Γ. These generalize the results of Stein and Weiss for p = 2 ([55]), as well as those of Qian et al for 1 ≤ p ≤ ∞ but only for dimension one ([44]). The obtained results in this aspect amounts to establishing the Paley-Wiener Theorems in higher dimensions for 1 ≤ p ≤ ∞. As related results we in this part prove the Cauchy and Poisson integral representation formulas for all p ∈ [1, ∞]. 2. We are to develop the decomposition of L p (R n ) functions, 0 < p < 1, into sums of Hardy space functions on tubes of the same indices. In the one-dimensional case, we know that L p (R) is identical with the direct sum of the Hardy spaces H+(R) and H−(R) for 1 < p < ∞ ([41] [44] [23]). That is, for any function f(x) ∈ L p (R), 1 < p < ∞, f(x) can be uniquely written as the sum of two functions f+ and f−, where f+ and f− are the boundary limits of functions in the Hardy spaces in the upper-half and lower-half planes, respectively. For higher dimensions analogous results for the range 1 < p < ∞ hold, that can be obtained by using the Fourier characterization results of the Hardy spaces developed in the first aspect, as well as be found in some literature([38]). Counterpart results, but lack of uniqueness, for 0 < p < 1 in the one-dimensional case are recently obtained by Deng and Qian in [16]. The second aspect of this thesis is to extend the L p spaces decomposition to the index range 0 < p < 1 in higher dimensions. In order to do so we need to establish rational approximation for higher dimensions. We also concern uniqueness issue of such higher dimensional decompositions. It turns out that, like in the one-dimensional case, the obtained decompositions are not unique either, and the corresponding decomposition of the spaces are not direct sums. 3. We consider the Hardy space decompositions of L p (∂D), 0 < p ≤ 1, where D stands for the open unit disc, and ∂D is its boundary. Hardy spaces decompositions for L p (∂D) and L p (R) for 1 ≤ p ≤ ∞ are, as classical results, available in the literature ([23], [8]). For 1 ≤ p ≤ ∞ the basic tools are the Plemelj formula, or alternatively the Fourier spectrum characterization of the disc Hardy spaces, and the boundedness of the Hilbert transformation. For 0 < p ≤ 1, neither on the real line, nor on the unit circle, a Plemelj formula is available. Also, there are no Fourier series, and nor Hilbert transformation are available for p < 1. Suggested by the recent paper of Deng and Qian on Hardy spaces decomposition of L p (R), 0 < p < 1, we achieve the same type decompositions on the unit circle by using rational approximations. The work on the unit circle exposes the particular features for compact domains. The methodology is adaptable to higher dimensions.