same issue here, I'm using ubuntu 14.10 64bit

I have tried install some :i386 libs, but I failed to install libgl1-mesa-glx:i386

sudo apt-get install -f libgl1-mesa-glx:i386
Reading package lists... Done
Building dependency tree       
Reading state information... Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies:
 libgl1-mesa-glx:i386 : Depends: libxext6:i386 but it is not going to be installed
                        Depends: libxxf86vm1:i386 but it is not going to be installed
E: Unable to correct problems, you have held broken packages.

I tried install libxext6:i386, and failed:

sudo apt-get install -f libxext6:i386
Reading package lists... Done
Building dependency tree       
Reading state information... Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies:
 gnome-icon-theme : Depends: libgtk-3-bin
                    Depends: humanity-icon-theme but it is not going to be installed or
                             gnome-icon-theme-full but it is not going to be installed
 ibus : Depends: gir1.2-gtk-3.0 (>= 3.8.5) but it is not going to be installed
        Depends: python-notify but it is not going to be installed
        Depends: libgtk2.0-0 (>= 2.24.5-4) but it is not going to be installed
 libcheese-gtk23 : Depends: libcheese7 (>= 3.4.0) but it is not going to be installed
                   Depends: libclutter-1.0-0 (>= 1.16) but it is not going to be installed
                   Depends: libclutter-gtk-1.0-0 (>= 0.91.8) but it is not going to be installed
                   Depends: libcogl20 (>= 1.17.4) but it is not going to be installed
                   Depends: gstreamer1.0-clutter but it is not going to be installed
 libgl1-mesa-glx : Depends: libxxf86vm1 but it is not going to be installed
 libgtk-3-0 : Depends: libxinerama1 but it is not going to be installed
              Depends: libxrandr2 (>= 2:1.2.99.3) but it is not going to be installed
              Recommends: libgtk-3-bin
 libqt5gui5 : Depends: libgtk2.0-0 (>= 2.24.0) but it is not going to be installed
 libxext6 : Breaks: libxext6:i386 (!= 2:1.3.2-1ubuntu0.1) but 2:1.3.2-1 is to be installed
 libxext6:i386 : Breaks: libxext6 (!= 2:1.3.2-1) but 2:1.3.2-1ubuntu0.1 is to be installed
 system-config-printer-gnome : Depends: gir1.2-gtk-3.0 but it is not going to be installed
E: Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.

Running 14.04.3 on a fresh install and I have this issue.

Every time I launch steam I am pestered to install some i386 libraries for Mesa. It never finds them. I thought the repo should have a directory listing... Is this correct?

http://repo.steampowered.com/steam/dists/precise/steam/binary-amd64/Packages

I am experiencing a similar issue, also on 14.04.3. :

Reading package lists... Done
Building dependency tree       
Reading state information... Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies:
 steam:i386 : Depends: libgl1-mesa-dri:i386
              Depends: libgl1-mesa-glx:i386
E: Unable to correct problems, you have held broken packages.

I tried:

sudo apt-get -f install

But it didn't resolve anything. Tried to resolve it with aptitude, which suggests to remove pretty much all installed packages. So I tried to install all unmet dependencies which led to:

sudo apt-get install libgl1-mesa-glx-lts-vivid:i386 libgl1-mesa-dri-lts-vivid:i386 libdrm2:i386 libglapi-mesa-lts-vivid:i386 libdrm-intel1:i386 libdrm-nouveau2:i386 libdrm-radeon1:i386

and this outputs:

Reading package lists... Done
Building dependency tree       
Reading state information... Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies:
 libglapi-mesa-lts-vivid:i386 : Conflicts: libglapi-mesa
E: Unable to correct problems, you have held broken packages.

So I ran

dpkg --get-selections | grep hold

which gives no results. I have googled around for 3 hours now and I can't seem to resolve the issue. Any ideas?

I just
sudo apt-get install steam -y

@Goddard Yeah I tried that, it did not help. I finally found out why my packages were broken. I had previously added the xorg-edgers PPA and installed the intel as well as the nvidia drivers from there, but at some point I must have removed the PPA without purging it. After readding it, apt-get was able to install the missing dependencies and all is well.

Similar problem here. I'm running a freshly installed 64bit Kubuntu 14.04.3 with an NVidia GeForce GTX 960 and nvidia-358.16 drivers. I had a hard time to get the hardware 3D acceleration running and the Steam client was one of the problems.

It is now working, but when I start the Steam client, it says:

Steam needs to install these additional packages: 
    libgl1-mesa-dri:i386, libgl1-mesa-glx:i386

However, when I do this, the 3D acceleration is gone again (at best) or the Xserver won't even start at all.
I do abort this prompt now every time (CTRL+C) and Steam starts and works without any problems.

Please install these packages before you will start try to install steam package:
sudo apt-get install
libc6:i386
libgl1-mesa-dri-lts-vivid:i386
libgl1-mesa-glx-lts-vivid:i386

This is workaround, but you should focus on fix this issue guys, 'casue it is really annoying and unprofessional.

Like I said, for me, installing these packages will break my system in that the 3D acceleration no longer works.

Today I found a solution about this problem. When I was searching, found that " sudo apt-get install libgl1-mesa-glx-lts-utopic:i386 ". And I tried like that : sudo apt-get install libgl1-mesa-glx-lts-trusty:i386 (for Ubuntu 14.04 LTS). It worked for me. Hope it will work for you...