Most linux distributions provide recent binary packages of Kdenlive that can be installed from your Package Manager. However, in some cases you can find more recent versions in private repositories.
This page at the Kdenlive home has instructions on how to source these more recent versions, depending on which flavour of Linux you run. Instructions for Debian, Fedora, Gentoo, OpenSUSE, Slackware and Ubuntu are available.
As of May 2015, obtaining and installing the latest version of Kdenlive has gotten a little complicated. Development of Kdenlive up to this point has used a technology known as KDE Frameworks 4. And the last stable release using this technology is version 0.9.10, released 25th September 2014.
To use this you need to have recent versions of frei0r, Kdenlive , mlt and libvidstab, e.g.:
As of June 2015, this PPA has version 0.9.10 of Kdenlive , 1.4.0 of Frei0r and 0.9.3 of melt.
Development of Kdenlive has now switched to a new version of the KDE Frameworks called KF5 (KDE Frameworks 5). To install versions of Kdenlive using this underlying technology, you need to be on a Linux distribution which supports KF5. Ubuntu and its derivatives versioned 15.04 and higher support KF5. It is not possible to install KF5 on distributions earlier than 15.04 (except by chrooting your system).
Versions using KF5 are distributed by the Kubuntu-CI Team. The sunab ppa (see below) now also distributes KF5 versions.
Version numbers of the KF5 flavour start at 15.04.0.
This author recommends using the sunab ppa to install kdenlive rather than the Kubuntu-CI Team because the Kubuntu-CI Team PPA has many other packages in it which will attempt to uprade all sorts of things on your system - with the potential of breaking it.
The switch between 0.9.10 and 15.04.0 is quite major with respect to the development work required. Testing of the new 15.04.0 version has been limited. For critical work I would recommend not upgrading to 15.04.0 unless you have a good backout strategy. It seems that the standard 14.04 Ubuntu repositories have a version 0.9.10 of Kdenlive running against KDE Frameworks 4 so you might be able to back out from 15.04.x to 0.9.10 by purging the ppa version of kdenlive and re-installing the one in the standard repos.
Debian project has been shipping Kdenlive packages since the "squeeze" (6.0) release. However, to benefit from recent updates and bugfixes, you might consider upgrading to a "testing" release or even "sid".
Once your package management tool is pointed at an appropriate release, a simple
apt-get install kdenlive should then work.
Sunab's PPA (see below) is not recommended for Debian because Debian uses a different lib layout for multiarch (reference: this post from vpinon).
Ubuntu also has offered Kdenlive since "gutsy" (7.10). However, to benefit from recent updates and bugfixes, you should consider upgrading to the latest release.
Sunab offers a stable release of Kdenlive (KF 5 version 15.04.x) from this PPA.
He maintains a ppa with the previous release here. As at June 2015 it has the ver 0.9.10 of Kdenlive , 1.4.0 of Frei0r and 0.9.3 of melt
Not all the packages in the ppa are necessary. Install or upgrade the following to get a 15.04 release (version numbers as at 13 July 2015)
Note that Sunab's PPA is built against the avconv multimedia libraries which Ubuntu/Lubuntu/Kubuntu et al. ship with. It is not built against the "alternative" ffmpeg multimedia libraries that other distros might have. So using this PPA for non-avconv-utilizing distributions might be problematic. (Thanks avconv/ffmpeg schism !!!!).
RPM packages are not yet maintained in an official branch, so you must go through an unofficial repository such as RPM Fusion or packman. Follow the site's recommendations to make them available and end with
yum install kdenlive.
Building scripts are ready for up-to-date systems, so run respectively
emerge kdenlive or
pacman -S kdenlive or
pkg_add kdenlive, etc.
There is also the kdenlive on win project on SourceForge which "consists of an Ubuntu VirtualBox image that is preconfigured to run Kdenlive". The project was last updated 2012-08-09.
Kdenlive and MLT can compile and run under Mac OS X. Packages are available from the MacPorts project.
MacPorts is a source-based system — there is not a binary app bundle for Kdenlive. Therefore, Kdenlive and all of its numerous dependencies, including multimedia libraries, KDE, and Qt, must be compiled. This can take a long time and much disk space! Furthermore, it is not unusual for something not to build correctly; it is definitely not something for the novice, impatient, or "faint of heart".
For more details see this.
You may have some success getting support for the MacPort of Kdenlive on the Mac Ports forum on MacOS Forge.
If you want to test latest committed code or your personal patches, you will have to build Kdenlive (and probably MLT) on your own.
You can use your distribution's package building procedure to use its software management system to install/upgrade/remove the binaries and data, and eventually share your builds (and even contribute to package maintenance - refer to the respective distribution manual).
If you prefer you can build & install Kdenlive to a local area (preferably not /usr, but rather /usr/local or $HOME/my_local_builds/kdenlive-last-release or similar). It is then recommended to use the build script  (Follow the instructions under "show kdenlive". These instructions will build Kdenlive and its dependancies [e.g. melt, ffmpeg] in a "sandbox").
Olivier Banus (AKA Sunab) provides a personal package archive (PPA) of the latest development trees of Frei0r-plugins, MLT and Kdenlive. For details on how to install Kdenlive using this bleeding edge source see launchpad.net - kdenlive-git (unstable).Testing only!
If you need to back out a release to the previous release then:
Dan Dennedy provides automatic daily builds that contain the latest git versions of Kdenlive, MLT, Frei0r, FFmpeg and some other important libraries. Note: It looks like the daily builds have not switched over to the ver 15.04 branch. So this will not give you the bleeding edge code.
You can test this packages without changing your system. Just download the package and extract it to a folder of your choice. Inside that folder is a "start-kdenlive" executable that will start for you the latest Kdenlive.
Execute this by typing
./start-kdenlive in a terminal.
Daily builds are currently provided for Ubuntu (should be compatible with OpenSuse 12 and Debian 7) and Fedora.
See the daily builds rss feed which provides links to the builds for the above mentioned distributions.
Be sure to choose the correct download for your system.
kdenlive-fedora17-x86_64-YYYYMMDD.tar.bz2 is for 64bit Fedora
kdenlive-ubuntu12.04-x86_64-YYYYMMDD.tar.bz2 is for 64bit Ubuntu - (release 12.04 or higher) or other Debian based distros
kdenlive-ubuntu14.04-x86-YYYYMMDD.tar.bz2 (no 64 in the name ) is for 32bit Ubuntu - (release 14.04 or higher) or other Debian based distros
Daily builds are often an easy solution if you are having troubles with your instal and can be preferable to builing your own because you don't need to install a build environment (which can be problematic on old distro).
Daily builds are likely to work even if your distro doesn't match the reference ones, as these references are older (with ascendant compatibility libs) and the important libs are embedded - it normally works without problem.
The potential downside is that the daily builds are built from the unstable development stream and may contain bugs. On the plus side your daily build version and your main version of Kdenlive can live simultaneously on your machine.
See also Kdenlive homepage
ENABLE_SWFDEC=0in the config variables of the script. Modify the INSTALL_DIR in the script to something like
INSTALL_DIR="$HOME/my_local_builds/kdenlive-last-release"to make it match where you want this local build to install.