This monograph is devoted to the study of the weighted Bergman space A ω p A^p_\omega of the unit disc D \mathbb {D} that is induced by a radial continuous weight ω \omega satisfying (†) lim r → 1 − ∫ r 1 ω ( s ) d s ω ( r ) ( 1 − r ) = ∞ . \begin{equation} \lim _{r\to 1^-}\frac {\int _r^1\omega (s)\,ds}{\omega (r)(1-r)}=\infty .\tag {†} \end{equation} Every such A ω p A^p_\omega lies between the Hardy space H p H^p and every classical weighted Bergman space A α p A^p_\alpha . Even if it is well known that H p H^p is the limit of A α p A^p_\alpha , as α → − 1 \alpha \to -1 , in many respects, it is shown that A ω p A^p_\omega lies “closer” to H p H^p than any A α p A^p_\alpha , and that several finer function-theoretic properties of A α p A^p_\alpha do not carry over to A ω p A^p_\omega . As to concrete objects to be studied, positive Borel measures μ \mu on D \mathbb {D} such that A ω p ⊂ L q ( μ ) A^p_\omega \subset L^q(\mu ) , 0 > p ≤ q > ∞ 0>p\le q>\infty , are characterized in terms of a neat geometric condition involving Carleson squares. These measures are shown to coincide with those for which a Hörmander-type maximal function from L ω p L^p_\omega to L q ( μ ) L^q(\mu ) is bounded. It is also proved that each f ∈ A ω p f\in A^p_\omega can be represented in the form f = f 1 ⋅ f 2 f=f_1\cdot f_2 , where f 1 ∈ A ω p 1 f_1\in A^{p_1}_\omega , f 2 ∈ A ω p 2 f_2\in A^{p_2}_\omega and 1 p 1 + 1 p 2 = 1 p \frac {1}{p_1}+ \frac {1}{p_2}=\frac {1}{p} . Because of the tricky nature of A ω p A^p_\omega several new concepts are introduced. In particular, the use of a certain equivalent norm involving a square area function and a non-tangential maximal function related to lens type regions with vertexes at points in D \mathbb {D} , gives raise to a some what new approach to the study of the integral operator \[ T g ( f ) ( z ) = ∫ 0 z f ( ζ ) g ′ ( ζ ) d ζ . T_g(f)(z)=\int _{0}^{z}f(\zeta )\,g’(\zeta )\,d\zeta . \] This study reveals the fact that T g : A ω p → A ω p T_g:A^p_\omega \to A^p_\omega is bounded if and only if g g belongs to a certain space of analytic functions that is not conformally invariant. The lack of this invariance is one of the things that cause difficulties in the proof, leading the above-mentioned new concepts, and thus further illustrates the significant difference between A ω p A^p_\omega and the standard weighted Bergman space A α p A^p_\alpha . The symbols g g for which T g T_g belongs to the Schatten p p -class S p ( A ω 2 ) \mathcal {S}_p(A^2_\omega ) are also described. Furthermore, techniques developed are applied to the study of the growth and the oscillation of analytic solutions of (linear) differential equations.