Not with the naked eye as the sunlight reflected by Earth makes that difficult, but we do have deep space telescopes, don't we?

I don't know much on this topic, but if we are able to spot planets that are light years away, then what's the problem with detecting objects that are gonna crash into us in the next 1000 years?

The only problem I can understand is that the sky is a pretty big place, but surely we also have a lot of time to detect such objects.

Planets outside of the solar system are not detected using light from the body itself. They are detected by their gravitational pull on their host star (i.e. tides; causing a slight twinkle in the light emitted by the star), or a transit (eclipse) of the star (causing the star to become slightly dim for a period). In both cases the telescope is looking at stars, and isn't fixated on one star but rather a field of stars, waiting for one to show activity.

Never mind the fact that even if we have a telescope that big, the issue is that these telescopes show a microscopically small field of view. There's a whole sphere that would need to be observed, and repeatedly (since the bodies move).

>The only problem I can understand is that the sky is a pretty big place, but surely we also have a lot of time to detect such objects.

I'm no expert, but as I understand it, the kind of objects we would be worried about are extremely hard to detect as they are dark, cold and relatively tiny.

At the same time, our "eyes" can only look at tiny patches of sky, generally through the distortion of our atmosphere for only a portion of each day with parts of the sky only visible part of the year. Furthermore, we generally have those eyes trained to other, known targets - stars, galaxies, nebulae and all that.

Consider that we didn't spot Apophis [1], which is set to come extremely close, until 2004 even though it's on our lawn [2] in terms of distance.

There is an effort [3] or two [4] afoot related to tracking asteroids and a new DARPA [5] program for tracking our own local junk.

1: http://en.wikipedia.org/wiki/99942_Apophis

2: http://www.howcloseisapophis.com/

3: http://www.skyandtelescope.com/news/160839035.html

4: http://www.planetaryresources.com/mission/

5: http://www.spaceviewnetwork.com/

We detect these planets light years away when they go between us and some light source, or when they change orbit of some light source (thanks to gravity). If the body is cold and the background is black we won't see it until it's close to our sun.