On Ubuntu (Debian distro) the easiest way is probably to run:

sudo apt install heif-gdk-pixbuf

For Fedora and other RPM-based distros use:

sudo dnf install libheif

After this, e.g. Eye of Gnome eog image.heic will display your image.

Ubuntu 22.04 quick start

• install heif-gdk-pixbuf for support in at least eog (Eye of GNOME, GNOME Image Viewer).
• install heif-thumbnailer for file manager (nautilus, nemo) thumbnails.
• The available versions of Geeqie, ImageMagick, gThumb and many more (see below) have built-in support.

Official packages

Starting with Ubuntu 20.04: At least for eog (Eye of GNOME, GNOME Image Viewer), you have to install the package heif-gdk-pixbuf (no PPA needed). In a terminal run:

sudo apt install heif-gdk-pixbuf

This will install libheif1 and libde265-0 along.

Apparently KDE distros with KDE Frameworks 5.80 have added support for HEIF and HEIC Image Formats to all KDE Apps (news, KDE announcement, MR), but the support flag is not enabled yet in the default (K)ubuntu packages (ubuntu bug).

Programs/Apps that support HEIF via libheif1 include:

• GNOME Image Viewer (Eye of GNOME - eog) is updated in Ubuntu 19.10 and later. (Thanks seanlano!) - if it does not work for you, see PPA tip below.
• gThumb Image Viewer & Organizer 3.11.4+, in ≥ Ubuntu 22.04 (for older verstions see below)
• Darktable, free open source photography application and raw developer (3.8+)
• GPicView (0.2.5+), can be installed from the Software Center or directly from the project page. Thank you, adasiko.
• digiKam (6.4+)
• GIMP (2.10.2+)
• Krita (4.2+)
• gImageReader (Ubuntu 20.04+)
• Kodi (free and open-source media player software application) with Add-on:HEIF image decoder
• xviewer (based on Eye of GNOME) with heif-gdk-pixbuf
• Shotwell (0.32+)
• Geeqie (1.6+) in Ubuntu 22.04
• heif-thumbnailer - a thumbnailer for HEIF images that can be used by Nautilus and Nemo.
• libheif-examples – provides command-line utilities: heif-convert and heif-enc.

See How best to search for dependencies?

Programs/Apps to CONVERT HEIF/HEIC-pictures:
Most of the above mentioned apps support to convert a HEIC-photo, e.g. eog (Eye of GNOME, GIMP, Krita, GPicView.
Open the photo via one of these image viewer/editor, then select Export to JPG (or another image format like PNG) or Save as (choose a different image format then HEIC).

For batch conversion on the command-line, use heif-convert or ImageMagick's convert.

Unofficial Packages and older Ubuntu versions

If one is willing to install a Flatpak, Snap app, or PPA:

• GNOME Image Viewer (Eye of GNOME - eog) can handle HEIF pictures with an updated gdk-pixbuf plugin which can be installed/updated with the above mentioned libheif-PPA on Ubuntu 14.04, 16.04, and 18.04. In a terminal run:

  sudo add-apt-repository ppa:strukturag/libheif
  

  Install the package heif-gdk-pixbuf. In a terminal run:

  sudo apt install heif-gdk-pixbuf
  

• gThumb Image Viewer & Organizer has an unofficial PPA for Ubuntu 18.04, Ubuntu 20.04, Ubuntu 21.04, Ubuntu 21.10, Linux Mint 20 and derivatives:

  sudo add-apt-repository ppa:ubuntuhandbook1/apps  
  sudo apt update  
  sudo apt install gthumb 
  

• Geeqie 1.6 has HEIF support. It is available in 22.04 ; it may be installed from AppImage, or on ubuntu 18.04 and 20.04 from PPA.

• ImageMagick 6.9 with HEIF support is available on ubuntu 22.04 ; on ubuntu 18.04 and 20.04 from bleeding edge PPA which also provides an updated libheif1 (1.12 instead of 1.6 as of October'21). If the colors seem wrong/inverted, you may need to convert from YCbCr with e.g. display -colorspace sRGB (source: github)

• Qt image plugin wraps libheif. It was created by jakar, and may be installed on ubuntu 14.04 to 19.04 from PPA. On more recent versions you may instead rebuild kimageformats as per bug 1951278.

Ubuntu 19.10
There are some prominent bugs in the shipped version of libheif1, so update to the latest version available from the PPA libheif by “struktur AG” team:
In a terminal: sudo add-apt-repository ppa:strukturag/libheif

Upgrade of packages (Sept 2020)
You also need to add the PPA strukturag/libde265 to get the correct dependencies:

sudo add-apt-repository ppa:strukturag/libde265  
sudo apt update  
sudo apt upgrade  

(If you get the error add-apt-repository: not found, make sure software-properties-common is installed, e.g. sudo apt install software-properties-common)

Those who do not wish to install a PPA may download the packages directly from the PPA webpages to install (dpkg -i) or recompile. Packages installed this way will not be automatically updated.

Online Tools

• Google Photos and Dropbox both support HEIF.

• There are other online converters (e.g. HEICtoJPEG), but check their privacy policies before use.

Miscellaneous

• ImageMagick, as of 7.0.7-22, can be compiled with the --with-libheif flag. One-liner build with IMEI. (The pre-built AppImage does not support heic yet.)

• tifig is described as "a fast HEIF image converter aimed at thumbnailing". However, it must currently be compiled from source.

• Nokia HEIF Info and Source Code. However, Apple's implementation differs somewhat.

• Windows 10 Build 17623 includes support for HEIF in the Photos app. However, the following may need to be installed before use:

  • HEIF Image Extensions

  • HEVC Video Extensions or HEVC Video Extensions from Device Manufacturer

I found that Converseen version 0.9.9.0 was able to convert an HEIC file downloaded from iphone to jpeg easily without any modification. It worked on Debian 11 Bullseye and assume that it should work on the current Ubuntu version. You may install it from repo using sudo apt install converseen

I hope it helps.

Convert them:

https://www.wilderssecurity.com/threads/iphone-how-to-convert-your-heic-photos.400574/

with this tool:

https://github.com/monostream/tifig

Here is the script to automate it for whole folders:

#!/bin/bash

for f in *; do

tifig -v -p "$f" "${f%.*}.jpg"

done

Then maybe set you iphone not to use this format.