Various rumours and discussions about whether The Amazing Spider-Man 3 was going to happen have been circulating for a while. Apparent leaks show Marvel Studios wanting to take over the character and remove Andrew Garfield with Sony keeping distribution rights. This seems to be down to the fact that Sony has been struggling to revive the franchise after The Amazing Spider-Man 2 didn't perform as well as hoped. Now though a recent casting call seems to suggest that The Amazing Spider-Man 3 is happening.
Bill Beckman Casting sent out a casting call looking for extras. This confirms that Sony is the studio producing. Plus it lists Andrew Garfield and Dane DeHaan and has this description "Peter Parker/Spider-man faces a new challenge to protect New York City". So it's pretty clear that Marvel Studios won't be involved and Sony Pictures is pushing on with a sequel. Casting calls usually only happen when shooting is just about to start, so it looks like things may be locked into place.