The one thing I've learned from countless WW2 documentaries is that allied code breaking was insanely helpful to the war effort.
I'm sure axis powers were well aware of how useful it would be to read allied radio traffic, but did they actually make any serious attempts at this? Did they succeed?
Bonus question: what about other countries, which were not so clearly on either side? Like Finland and Sweden, for example.
I don't know much about other countries' efforts but Finnish cryptanalysts were fairly successful. Here's a good summary in English:
http://chris-intel-corner.blogspot.fi/2012/09/the-finnish-cryptologic-service-in-wwii.html
TL;DR: Finnish intelligence decoded many Soviet military codes fairly regularly (and contributed considerably to war effort), and because of a capture of almost intact one-time pad could read many high-level messages. They also breached U.S. M-138-A diplomatic code among many others (e.g. Yugoslav, Turkey). These were used to gain insights into political developments. There was a lot of cooperation, first with British and then with the Germans, extensive cooperation with the Swedish intelligence, and important connections to Japan and other warring countries. After the war, lots of personnel fled to Sweden and plenty of material ended up in U.S. hands.
The German Navy's SIGINT group, the B-Dienst, had a large amount of success at cracking Allied naval codes in the 1930s, and the early 1940s. During this period, the RN and USN mostly used code books, which were simpler to use, produce and distribute, but also easy to break. At the start of the war, the B-Dienst could read enough of the RN's messages to be aware of the locations of every major warship. They could also break the BAMS code used to talk to merchant ships, aiding U-boat attacks on convoys. Until the USN switched to cipher machines in April 1942, the B-Dienst could read many of it's messages, contributing to the success of Operation Drumbeat. As new codes were introduced by the British, the B-Dienst was able to keep up, but was not always able to produce workable intelligence - they broke ~80% of messages in Naval Cipher No 3 that they intercepted, but only about 10% were broken quickly enough to be useful.
Source:
German Naval Codebreakers, Jak Mallmann Showell, US Naval Institute Press, 2003
The Japanese had three dedicated code breaking units both before the war and during. They had some success with various codes. But their best success was against the US M-209 encryption units.
By 1944 these were used by frontline units in the army and army air force. The Japanese cracked the system. But the system used a double cypher that was changed daily, and had to be cracked daily. So while there were days that the Japanese were reading the messages, there were also days that they weren't.
Japanese Intelligence in World War II’ by Ken Kotani