About a month ago I made the decision to start watching Andrei Tarkovsky's films.  I had heard about this Russian master film-maker before, but really had no idea where to begin in his catalogue.  I chose his most famous two to start with, "Stalker" and "Solaris".  I'm hooked.  When you watch his movies, they instantly get under your skin, but they don't stop there.  They leak into your chest and your head (even your soul perhaps?).  They leave you feeling completely shaken and full of questions, desperate to talk with someone about the experience.  They are unlike anything I have ever seen.

Stalker (1979):
Stalker is Tarkovsky's masterpiece. It is a film both profoundly simple, and simply profound (clever...?).  The film follows a central figure "The Stalker" as he guides "Writer" and "Professor" through "The Zone".  The Stalker is a man of the land, a man who knows how to guide people through a stretch of abandoned Russian landscape rumored to hold both the secrets to life, but also terrifying and deadly pitfalls and traps.  The film unfolds incredibly slowly, with Tarkovsky's trademark never ending shots, and it is impossible to shake the sense of dread and loneliness that permeates the film from beginning to end.  I want to say more, but its incredibly hard to describe a film like this.  This youtube trailer is but a taste:  
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Its the sound and the music that really stick with you.

Solaris (1972):
Solaris is often compared to 2001, as both are science fiction films about mankind's future experiences at both the edge of space and comprehension.  The film was actually remade, with George Clooney in the lead, a few years ago.  I haven't seen this version but I hear it wasn't bad.

Unlike 2001, Solaris is far more intimate and introspective.  It follows a character named Kris, who travels to a station orbiting a planet covered in a global ocean to find out why the 3 crew members have stopped their research.  He quickly learns the mystery of Solaris: that the planet is alive and manifests itself in the form of people from the crews past memories.  Kris is forced to re-live his most painful memory, that of his deceased wife, while trying to work with the remaining crew (who are equally crazy).  This is unfortunately the best taste I could find:
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These films are long (2.5-3 hours), and both extremely slow moving and dark, but well worth the time.