Arrhenius acids and bases

According to the Arrhenius definition, an acid is any substance, which when dissolved in water, tends to increase the amount of ${\rm H}^+$. An example is HCl:

$${\rm HCl}(g)\longrightarrow {\rm H}^+(aq) + {\rm Cl}^-(aq)$$

An Arrhenius base is any substance, which when dissolved in water, tends to increase the amount of OH$^-$. An example is NaOH:

$${\rm NaOH}(s) \longrightarrow {\rm Na}^+(aq) + {\rm OH}^-$$

These definitions are correct but not general enough to include the wide range of acid and base substances which are known to exist. In addition, they rely on the use of water as a solvent, which is also too narrow.