Six new cobalt(ii) metal-organic frameworks, {[Co1.5(TTPA)(BTC)(H2O)]2·13H2O}n (1), [Co(TTPA)(PA)]n (2), {[Co(TTPA)(BDA)0.5(NO3)]·3H2O}n (3), {[Co2(TTPA)3(OBA)2(H2O)3]·2CH3CN·4H2O}n (4), {[Co(TTPA)(AIP)(H2O)]·2H2O}n (5), and {[Co(TTPA)(MIP)(H2O)]·2H2O}n (6), have been prepared by the self-assembly of a tris(4-(1H-1,2,4-triazol-1-yl)phenyl)amine (TTPA) ligand with different aromatic carboxylate auxiliary ligands (H3BTC = 1,3,5-benzenetricarboxylic acid, H2PA = phthalic acid, H2BDA = (1,1'-biphenyl)-4,4'-dicarboxylic acid, H2OBA = 4,4'-oxydibenzoic acid, H2AIP = 5-aminoisophthalic acid, H2MIP = 5-methylisophthalic acid) and cobalt salts. Their structures have been characterized by infrared spectroscopy, elemental analysis, single crystal X-ray analysis, and powder X-ray diffraction. Complex 1 is an unusual 4-nodal (3,3,4,8)-connected three-dimensional (3D) new topological net with the point symbol of (4·6·8)4(4(4)·6(7)·8(15)·10(2))(6(2)·8(4)). Complex 2 has a 2-fold interpenetrating 3D dia framework. Complex 3 displays a rare binodal (3,4)-connected 4-fold interpenetrating 3D architecture with a fsc-3,4-C2/c topology with the point symbol of (4·6·8)(4·6(2)·8(3)). Complex 4 shows two distinct two-dimensional (2D) layers with hcb topologies, which are further packed into a 3D structure by O-HO hydrogen bonding interactions. Both complexes 5 and 6 feature similar 2D sheets with sql topologies, which can be further packed into a 3D structure by O-HO hydrogen bonding interactions. Moreover, the thermal stability and UV-visible spectra of these complexes are also discussed in detail. Meanwhile, the variable-temperature magnetic susceptibility measurement of complex 1 reveals antiferromagnetic interactions between Co(ii) ions.