The Lewis theory of acids and bases states that acids act as electron pair acceptors and bases act as electron pair doners. This definition doesn't mention anything about the hydrogen atom at all, unlike the other definitions. It only talks about the transfer of electron pairs. To demonstrate this theory, consider the following example.
This is a reaction between ammonia (NH3) and boron trifluoride (BF3). Since there is no transfer of hydrogen atoms here, it is clear that this is a Lewis acid-base reaction. In this reaction, NH3 has a lone pair of electrons and BF3 has an incomplete octet, since boron doesn't have enough electrons around it to form an octet.
Figure 2. The Lewis structures of ammonia and boron trifluoride.
Because boron only has 6 electrons around it, it can hold 2 more. BF3can act as an acid and accept the pair of electrons from the nitrogen in NH3, which will then form a bond between the nitrogen and the boron.
Figure 3. The Lewis structure of H3NBF3, which resulted from the bond between nitrogen and boron.
This is considered an acid-base reaction where NH3 (base) is donating the pair of electrons to BF3. BF3 (acid) is accepting those electrons to form a new compound, H3NBF3.