Small weighted Bergman spaces

This paper is based on the course \lq\lq Weighted Hardy-Bergman spaces\rq\rq\, I delivered in the Summer School \lq\lq Complex and Harmonic Analysis and Related Topics\rq\rq at the Mekrij\"arvi research station of University of Eastern Finland, June $2014$. The main purpose of this survey is to present recent progress on the theory of Bergman spaces $A^p_\omega$, induced by radial weights $\omega$ satisfying the doubling property $\int_r^1\omega(s)\,ds\le C\int_{\frac{1+r}{2}}^1\omega(s)\,ds$.