The Tile Calorimeter (TileCal) is a sampling hadronic calorimeter covering the central region of the ATLAS experiment, with steel as absorber and plastic scintillators as active medium. The operation of High Luminosity LHC targets 4000 fb-1 of total integrated luminosity with an increase of a factor of five to the LHC nominal instantaneous luminosity. New electronics of the TileCal is needed to meet the requirements of a 1 MHz trigger, higher ambient radiation, and to ensure better performance under high pile-up conditions. Both the on- and off-detector TileCal electronics will be replaced during the shutdown after the end of the LHC operation. The modular front-end electronics feature radiation-tolerant commercial off-the-shelf components and redundant design to minimise single points of failure. The timing, control and communication interface with the off-detector electronics is implemented with modern Field Programmable Gate Arrays (FPGAs) and high speed fibre optic links running up to 9.6 Gb/s. The results of the extensive R&D programme for on- and off-detector systems, together with expected performance and results of beam tests with the electronics prototypes will be discussed. A demonstrator module which uses partly the upgrade electronics and is designed to be compatible with the current ATLAS read-out chain is in continuous operation since it was inserted in the TileCal in 2019.