We discuss experimental data, obtained on organic quasi-one-dimensional (TMTSF)2X (X=PF6 and ClO4) superconductors, which reveal unconventional nature of their superconducting phases. Our theoretical interpretation of the data supports a triplet scenario of superconductivity in (TMTSF)2PF6 material. In particular, we suggest a careful theoretical analysis of the recently measured anisotropic upper critical fields. On a basis of this analysis, we suggest that the triplet superconducting order parameter is described by a d-vector with da(k) ≠ 0 and db(k) = dc(k) = 0, which defines a spin part of the order parameter but does not uniquely define its orbital part, da(k). We consider two major possible orbital parts of this triplet order parameter, px-wave and py-wave ones. In the end of the chapter, we discuss in a brief a d-wave singlet scenario of superconductivity, which we consider as a possibility in (TMTSF)2ClO4 material.