We study the communication primitives of broadcasting (one-to-all communication) and gossiping (all-to-all communication) in known topology radio networks, i.e., where for each primitive the schedule of transmissions is precomputed in advance based on full knowledge about the size and the topology of the network. We show that gossiping can be completed in $O(D+\frac{\Delta\log n}{\log{\Delta}-\log{\log n}})$ time units in any radio network of size $n,$ diameter $D$, and maximum degree $\Delta = \Omega(\log n).$ This is an almost optimal schedule in the sense that there exists a radio network topology, specifically a $\Delta$-regular tree, in which the radio gossiping cannot be completed in less than $\Omega(D+\frac{\Delta\log n}{\log{\Delta}})$ units of time. Moreover, we show a $D+O(\frac{\log^3 n}{\log{\log n}})$ schedule for the broadcast task. Both our transmission schemes significantly improve upon the currently best known schedules in G\k{a}sieniec, Peleg, and Xin [PODC'05], i.e., a $O(D+\Delta\log n)$ time schedule for gossiping and a $D+O(\log^3 n)$ time schedule for broadcast. Our broadcasting schedule also improves, for large $D$, a very recent $O(D+\log^2 n)$ time broadcasting schedule by Kowalski and Pelc.