This monograph is devoted to the study of the weighted Bergman space $A^p_omega$ of the unit disc $mathbb{D}$ that is induced by a radial continuous weight $omega$ satisfying $lim_{rto 1^-}frac{int_r^1omega(s),ds}{omega(r)(1-r)}=infty.$ Every such $A^p_omega$ lies between the Hardy space $H^p$ and every classical weighted Bergman space $A^p_alpha$. Even if it is well known that $H^p$ is the limit of $A^p_alpha$, as $alphato-1$, in many respects, it is shown that $A^p_omega$ lies ``closer'' to $H^p$ than any $A^p_alpha$, and that several finer function-theoretic properties of $A^p_alpha$ do not carry over to $A^p_omega$.