Like OhtheRoses I knew from about 6 weeks. DD was a very alert baby who hardly slept (but didn't cry much either); but she didn't talk super-early or anything like that. It was more that she clearly had an amazing grasp of the underlying concepts behind things, and never seemed to need to practice anything. She had amazing fine motor skills and acute eyesight which helped, but you could see an amazingly precise intuition about things going on. At 5 weeks she realised that she could use a round O-ball toy to make the dangling bits on her baby gym swing, and set about hitting them from different sides to see them move. At two months she was passing toys from hand to hand and turning them round as she did so to see their shadows change shape on a wall. At three months she would turn the pages of a board book without prompting when we were reading to her. So we worked out pretty early that she was very bright, and it wasn't correlated to early talking or anything like that - in fact she had her "language explosion" at about 17 months. At nearly 4 she is close to being a free reader, and is amazingly sharp conceptually, though we are pretty relaxed about teaching her learned skills like maths. Those can always be picked up by a clever child.
It helps I think that both DH and I were what would now be called G&T when we were young (I also had an eidetic memory, like MaQueen's DD, which disappeared in my early twenties, oh how I miss it!) Though I was in mainstream school (and hated it), DH was accelerated in G&T programmes (and hated it). So we're very aware of the pitfalls that await very bright children, and we are aiming to get a balance between helping DD but making sure she is as happy and socially rounded as possible.
Always be aware, though, that as problems go it is a pretty enviable one. With love and family support, most very bright children end up forging their own path and doing fine even in the absence of tons of learning support. There is something to be said for allowing them to develop other skills and not focusing too much on their talents - learning how to work hard and persistently, to be a good friend and develop their artistic or physical skills as well as their academic skills. The worst that can happen really is that they slightly underachieve where they could have got if they had been hothoused at Westminster etc.; but they are unlikely not to succeed in life. I give thanks for having my bright DD when lots of friends have had heartache in their lives and their children - really, I can't worry too much about the finer details of what she's achieving when in the wider scheme of things it is a joy to have such a clever energetic young thing. I want her to enjoy stories, imaginative play, make friends, have interesting experiences, be bored, be a bit frustrated (it's really important for bright kids occasionally to be bored and frustrated). Please don't worry about what your 1 y o is doing, just enjoy being with them and interacting! Read them loads of stories; sing them songs! Much better than anything else at that age!