We introduce LTrack, a new tracking attack on LTE that allows an attacker to stealthily extract user devices' locations and permanent identifiers (IMSI). To remain stealthy, the localization of devices in LTrack is fully passive, relying on our new uplink/downlink sniffer. Our sniffer records both the times of arrival of LTE messages and the contents of the Timing Advance Commands, based on which LTrack calculates locations. LTrack is the first to show the feasibility of a passive localization in LTE through implementation on software-defined radio. Passive localization attacks reveal a user's location traces but can at best link these traces to a device's pseudonymous temporary identifier (TMSI), making tracking in dense areas or over a long time-period challenging. LTrack overcomes this challenge by introducing and implementing a new type of IMSI Catcher named IMSI Extractor. It extracts a device's IMSI and binds it to its current TMSI. Instead of relying on fake base stations like existing IMSI Catchers, which are detectable due to their continuous transmission, IMSI Extractor relies on our uplink/downlink sniffer enhanced with surgical message overshadowing. This makes our IMSI Extractor the stealthiest IMSI Catcher to date. We evaluate LTrack through a series of experiments and show that in line-of-sight conditions, the attacker can estimate the location of a phone with less than 6m error in 90% of the cases. We successfully tested our IMSI Extractor against a set of 17 modern smartphones connected to our industry-grade LTE testbed. We further validated our uplink/downlink sniffer and IMSI Extractor in a test facility of an operator.