Some disclosed suture securement devices comprise a flat, thin, generally planar body that has one or more suture engagement slots extending into the body from a perimeter inlet for receiving sutures laterally into the device. The slots can be resiliently widened to receive a suture and then released to clamp onto the suture. Some embodiments include two or more such suture engagement slots that independently receive and secure different sutures. Some embodiments include one or more locking tabs that have a closed position that retains the slots against widening and blocks the inlet to the slot. In some embodiment two or more slots are position next to each other on the same side of the device, while in other embodiments slots are positioned on opposite sides of the device. Other types of suture securement devices are also disclosed.