What are the biggest risks when downloading torrents? How likely are they to occur? How to avoid them?

malware,spyware,viruses,dialers,smitfrauds,trojans,child porn,ransomware and worse. scan file with an av and anti-malware first before opening.only torrent from reputable sites. my opinion on downloading torrents (like music) is if you download it for personal use it's ok but if you download it to mass produce and sell then it is illegal especially if it is copyrighted. 

No risk at all (or minimal) if downloading things like Linux distributions or other reliable software.

If downloading movies or music, there is a risk you can be tracked by the copyright owners, and your ISP notified (and your) to stop, else legal action will be taken,

So, the answer depends on "what" you are downloading via torrents. Some of it is 100% legal and safe, others are illegal or questionable and risky....

Torrents itself is not illegal, lots of distributions use it. It is the content one is downloading that determines the legality of it....

NEVER EVER download an executable from a torrent site.  Video or audio might work but make sure they are not executable.  Playable yes, executable NEVER.  The VAST majority of torrents are illegal.  Bad guys have no problem inserting malware into torrents that you shouldn't have in the first place.

Bittorrent itself is not illegal.  Your X-Box and PlayStation both use it for game updates, mainly to keep their servers from getting SLAMMED when the game vendor releases an update.  Picture it, 1 million 50 Gig updates.  That is 53.68 petabytes. (I use 1024 byte K's, not 1000 byte K's, known as a Kibibyte.  Your download speed from the ISP uses 1000 as do your hard drive sizes.  This is why your 1 TB drive is only 931 Gibibytes.  This can get very confusing so HD and others use Kibibyte for 1024 to separate it from Kilobyte at 1000 )

Can you imagine the bill they'd face?  Neither can I and that is why they torrent it and the updates start slow, then speed up as the system learns of more seeders.  Then you seed for others (ALWAYS do that!)

"downloading TORRENTS" is THEFT.

To avoid risks, Don't Do It.

Risks include virus/malware which can disable your computer, allow your computer to be "taken over" by criminal entities. Conversion of your personal data.