Various resources   state that the optimal method of charging a li-ion cell -- such as one found in a mobile phone -- is to charge at a constant current (usually <1C) until a certain voltage threshold is reached, then switch to charging at a constant voltage until the charging current drops to about 0.1C, at which point the battery is fully charged.
The exact target voltage differs slightly for different battery chemistries, but according to National Semiconductor (see figure 5) it's common to switch from CC to CV mode at 4.1V with the CC target voltage being 4.2V.
Many manufacturers sell inexpensive charge-controller ICs that take the guesswork out of charging li-ion cells, but they're not as widely available to hobbyists as, say, an LM317 voltage regulator. Additionally, they tend to be small, SMT devices that are not breadboard friendly.
Although both constant-current and constant-voltage charger circuits can be built with an LM317, a current-limited, constant-voltage circuit is particularly simple to build.
Is there a downside to charging at a constant voltage (say 4.2V) so long as the maximum current is limited to a reasonable value for the cell (say 0.5C)?
(In this context, consider "downside" to be more along the lines of "cell doesn't fully charge with CV-only, is damaged, explodes, steals your lunch money, burns house down, etc." and less along the lines of "CV-only charging takes longer", though that is also useful information to know.)
I ask because the internal charger circuit in my mobile phone died unexpectedly and, while waiting for a replacement phone, I constructed a CC-CV charger using an LM317 and successfully charged the battery. This was a bit messier than expected, and I was curious if a simple current-limited constant voltage circuit would also be suitable for occasional use.
Possible duplicate of this question, which was answered with "If you really want to charge battery from time to time - charging with resistor and constant voltage 4.2V or less will work and battery will not blow up if you choose proper resistor." and a caution to ensure it's OK for the cell to be charged at high current when empty.
Fortunately, charger ICs like the TP4056 as well as assembled PCBs with such a chip and the necessary connecting pins are available cheaply on eBay and other vendors. Since I needed to safely charge lithium batteries for other, unrelated, electronics projects I've purchased several of them and keep a few spares around just in case. They work very well.