In this paper we study the positive Borel measures μ on the unit disc $${\mathbb{D}}$$ in $${\mathbb{C}}$$ for which the Bloch space $$\mathcal{B}$$ is continuously included in $$L^p(d\mu)$$ , 0 1, the p-Bloch-Carleson measures are exactly those for which the Toeplitz operator $$T_\mu$$ , defined by $$T_\mu(f)(z) = \int_\mathbb{D} {\frac {f(w)} {(1-\bar{w}z)^2}} d\mu(w) (f \epsilon L^1(d\mu), z \epsilon {\mathbb{D}})$$ , maps continuously $$L^{p\prime}\,(d\mu)$$ into the Bergman space A 1, $$\frac {1} {p}\,+\,\frac {1}{p\prime}\,=\,1$$ . Furthermore, we prove that if p > 1, α >-1 and ω is a weight which satisfies the Bekolle-Bonami $$\mathcal{B}_{p,\alpha}$$ -condition, then the measure $$\mu_{\alpha,p}$$ defined by $$d\mu_{\alpha,p}(z) = {(1-|z|^2)}^{\alpha}\omega(z)dA(z)$$ is a p-Bloch-Carleson-measure. We also consider the Banach space $$H^{\infty}_{\rm log}$$ of those functions f which are analytic in $${\mathbb{D}}$$ and satisfy $$|f(z)| = O\left({\rm log} \frac {1} {1-|z|}\right)$$ , as $$|z| \rightarrow 1$$ . The Bloch space is contained in $$H^{\infty}_{\rm log}$$ . We describe the p-Carleson measures for $$H^{\infty}_{\rm log}$$ and study weighted composition operators and a class of integration operators acting in this space. We determine which of these operators map $$H^{\infty}_{\rm log}$$ continuously to the weighted Bergman space $$A^{p}_{\alpha} (p > 0, \alpha > -1) $$ and show that they are automatically compact.