A general summability method of Fourier series and Fourier transforms is given with the help of an integrable function θ having integrable Fourier transform. Under some weak conditions on θ we show that the maximal operator of the θ-means of a distribution is bounded from H p (T) to L p (T) (p 0 <p<∞) and is of weak type (1,1), where H p (T) is the classical Hardy space and p 0 <1 is depending only on θ. As a consequence we obtain that the θ-means of a function f∈L 1 (T) converge a.e. to f. For the endpoint p 0 we get that the maximal operator is of weak type (Hp 0 (T), Lp 0 (T)). Moreover, we prove that the θ-means are uniformly bounded on the spaces H p (T) whenever p 0 <p<∞ and are uniformly of weak type (Hp 0 (T), Hp 0 (T)). Thus, in the case f∈H p (T), the θ-means converge to f in H p (T) norm (p 0 <p<∞). The same results are proved for the conjugate θ-means and for Fourier transforms, too. Some special cases of the θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, Riemann, de La Vallée-Poussin, Rogosinski and Riesz summations.