LATAM is the name formed by the acquisition  and takeover by Chilean, Lan Airlines of Brazilian, TAM Airlines, which was completed on June 22, 2012. The full integration was not complete until 2018, when the name of TAM Airlines disappeared and all aircraft rebranded under the one name. To understand the history of the two airlines please refer to the detailed history article below by clicking their respective former logos:

The beginning of the cooperation between the two major airlines started  the 3 August  2010, when LAN (Chile) signed a non-binding agreement with Brazilian airline TAM Airlines to merge, and form the LATAM Airlines Group. The official agreement then took place two years later in 2012. The two airlines would still operate separately, despite being managed under a common executive group. Their aim would be to streamline their operations and benefit from each others strengths in their respective markets and become South America's biggest airline group. 

The group created the subsidiaries of LATAM Argentina, LATAM Brasil, LATAM Cargo Brasil, LATAM Chile, LATAM Cargo Chile, LATAM Express, LATAM Colombia, LATAM Cargo Colombia, LATAM Ecuador, LATAM Cargo Mexico, LATAM Paraguay and  LATAM Perú to be able to better administrate each airline in their respective countries. 

(This article focuses on the LATAM Chile and LATAM Brazil subsidiaries)

After the merger, a period of consolidation and harmonization started between the two airlines, the brazilian side  signed a firm contract with Airbus to acquire 37 additional aircraft. The order comprised twelve A319s, 16 A320s, three A321s and three A330s The deliveries were concluded by 2013. Also it announced that  LATAM Brazil would phased out 3 of the oldest Boeing 767 it operates; however, it later changed plans and decided to keep the aircraft, adding some more aircraft from LATAM Chile. They replaced the A330-200s. LATAM Brazil also received the first aircraft of the A320 family with Sharklets in April 2013.

LATAM Chile bought the Boeing 787 for its long haul routes such as Auckland, Sydney and European routes, replacing its Airbus A340-300s, that left the fleet in April 2015. The first Boeing 787-8 arrived in the Summer of 2012 whilst the -9 versions arrived in May 2016. 

Since May 5, 2016, LAN Airlines operating as LATAM Chile, then opened many routes during 2017, one of them being their longest flight in their history: Santiago to Melbourne, which started operating October 5 of that year. LATAM Chile received its first A350 in early 2016 as well as for LATAM Brazil.

In November 2017 LATAM Chile announced the opening of a direct air route to The Middle East. The route will operate with a flight departing from Santiago, Chile, make a stop in Sao Paulo, Brazil  and continue direct to Tel Aviv, Israel. The flights started from December 2018 with new Boeing 787-9 aircraft. In December 2017 the airline received their very first Airbus A320neo aircraft. 

History source: Wikipedia.org + LATAM.com