A remarkable distinction between boron and carbon hydrides lies in their extremely different bonding patterns and chemical reactivity, resulting in diverse areas of application. Particularly, carbon, characterized by classical two-center - two-electron bonds, gives rise to organic chemistry. In contrast, boron forms numerous exotic and non-intuitive compounds collectively called non-classical structures. It is reasonable to anticipate that other elements of Group 13 exhibit their own unusual bonding patterns; however, our knowledge of the hydride chemistry for other elements in Group 13 is much more limited, especially for the heaviest stable element, thallium. In this work, we performed a conformational analysis of Tl2 Hx and Tl3 Hy (x=0-6, y=0-5) series via Coalescence Kick global minimum search algorithm, DFT, and ab initio quantum chemistry methods; we investigated the bonding pattern using the AdNDP algorithm, thermodynamic stability, and stability toward electron detachment. All found global minimum structures are classified as non-classical structures featuring at least one multi-center bond.