I am using a Mio Fuse. You can probably get one for about £100 now.
As a heart rate monitor, it is generally pretty good. Seems fairly reliable and accurate for me, for both running and cycling. I've found it is a bit fussy for position, there is a particular part of my wrist where it works best. And it does need to be fairly tight.
But definitely more comfortable than a chest strap for me, I got horrible chafing from the last Garmin chest strap I used.
It can work as an activity monitor (or as a watch), but it is pretty basic. It is essentially just a step counter. Plus it will give numbers for distance and calories. I'm not sure about the accuracy for this, but it generally seems consistent. For running, the distance is usually about 10% less than my Garmin GPS watch.
It doesn't do any sleep monitoring, but Mio have promised that in a future update, don't know when.
It can connect to an app on your phone/tablet. Though the app seems fairly limited. It will let you view daily totals, and graphs for heart rate etc, but no way to export your data, or share it online etc. There have been some recent updates, so maybe it will improve.