It's a hauntingly beautiful film from the spiritually resonant musical score by Zbigniew Preisner to the ethereal lighting of cinematographer Slawomir Idziak.
The film is filled with images that will stick with me for a long time. As described in the commentary that accompanies the DVD, the film truly is a visual poem. In shots as simple as a teabag dancing in a glass of water to the more complicated compositions of a crowd swarming a stage, the film is filled with memorable moments.
I don't want to over-analyze the story, but thematically the film doesn't stray from Kieslowski's other works such as the Decalogue or Three Colors: Blue, White, Red. What struck me personally was the treatment of the human yearning for connection and meaning.
The story in simplest form is that of two women with twin souls, born on the same day to different mothers in different countries. They have similar experiences and intuit the existence of their counterpart. The Polish Veronica dies tragically young, and this creates an emptiness in the French Veronique that she feels deeply but can't explain to herself or those around her. It's a void that she tries to fill via romantic love, unsuccessfully if I interpret the ending correctly.
It does what every great film should -- leaves you with many questions, and touches you in ways you can't explain, just like Veronique.