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We characterize the Schatten class Toeplitz operators induced by a positiveBorel measure on the unit disc and the reproducing kernel of the Bergman space$A^2_\omega$, where $\omega$ is a radial weight satisfying the doublingproperty $\int_r^1\omega(s)\,ds\le C\int_{\frac{1+r}{2}}^1\omega(s)\,ds$. Byusing this, we describe the Schatten class composition operators. We alsodiscuss basic properties of composition operators acting from $A^p_\omega$ to$A^q_v$.

Trace class criteria for Toeplitz and composition operators on small Bergman spaces