Bi-directional cycleways are a tricky thing to get right, and they are often not appropriate in urban areas - on single-carriageway roads uni-directional cycleways on each side of the road should be the standard treatment.
This example in the Dutch town of Wageningen makes a good deal of sense, however. It runs parallel to a major N-class road (a British A-road) that passes through the town, and, importantly, is on the town centre side of this road. As the town centre is access-only for motor traffic, placing a bi-directional cycleway on this side of the road means that there are limited turning conflicts with motor traffic. A cycleway on the other side of the road - be it bi-directional, or uni-directional - would have to address conflicts with motor traffic turning into through-roads, as well as buses entering and exiting the town's bus station.
This bi-directional cycleway can be accessed from the town centre roads without needing to wait at any signals, and is of course well-separated from the motor traffic running on the main road. It's also smooth and wide; wide enough for three people to ride side-by-side on one side of the cycleway.