Nanorobots. What stops us from producing them yet? If we can already predicts accuratelly motion on molecular levels, what stops us from developing small robots to, for instance, navigate through our blood vessels looking for cancerous cells and destroying them? What are the predictions for when we will able to do this?
If someone is very interested on this goal and wants to participate on research, what should exactly he study? Programming, chemistry, physics, quantum mechanics? What could help at all? Books/resources welcome.
 A: I think the real answer is that when it comes to nanorobots, the materials we're using readily oxidise. Put them out of a vaccum and they're toast the instant they come into contact with the atomosphere. 
Biology manages to deal with this by using a different material set, and encapsulating everything pretty well so that the environment doesn't damage cell contents.
If you're seriously interested in learning this for biomedical uses, I think you'd be better served by reading up on biochemistry, new research has already printed DNA, coded by humans, to control artificial cells. These are probably what will be used in the body first. This and targeted viruses.
A: I don't believe that the materials we use for engineering on the macroscopic scales, gears and wheels and metals and so forth, will be appropriate for nano-devices. Macroscopic devices are usually built out of chemically homogenous materials, which do not afford flexibility in knowing where to cut and splice to design new shapes, or how to assemble the parts into a working machine. For example, when you build a mechanical gear out of cut metal, you need to mechanically make a precision measurement of length from an edge to know where to cut, and this is just difficult to do on a microscopic scale, because things jitter, and you can't see what you're doing.
So in order to engineer devices on a nano-scale, the easiest way is if you can sniff out where your parts are, by chemically labelling the positions. This is easiest in chemically completely inhomogenous materials, where every molecule and submolecule is unique and identifiable by its active chemical properties and chemical enviroment.
Designing such materials from scratch would give any chemist nightmares. Fortunately, nature already did it for us! We have bacteria and single-cell eukariotes, which are already doing all sorts of mechanical and chemical things with interchangable and engineerable parts. It is much more plausible that we will build biological devices on the scale of biological cells, with the ability to perform complex tasks which exhaust your intended domain.
Biological things are environmentally friendly.
Using biological materials has the added benefit of complete bio-degradability, and it is potentially complete environmentally cost-free. Science fiction authors like to imagine a human-designed super-bug, which accidentally outcompetes the natural ones, it is almost certain we would have to work pretty hard to come up with a bug designed from scratch which would even be able to survive in the world. We can a mouse that glows in the dark. You can just imagine how long that mouse would last in a forest.
It is very difficult to imagine an accidental super-bug, although it is possible that one might be designed on purpose, of course.
If you are very concerned about non-mixing of artificial and natural life, it is theoretically possible to make all artificial living things of opposite chirality molecules. Such molecules will be alien to life on earth, but chemically indistinguishable in a mirror image sense, so they work the same. The drawback, of course, is that you have to get all the chemicals from scratch, you can't eat biological things to extract spare parts. But once you build a mirror-image plant, and a mirror image nitrogen-fixing bacterium, etc, you can set up a mirror image plot of land, and nothing from the regular biological world can contaminate, because a jungle of opposite chirality life will seem to be a wasteland from the point of view of ordinary life.
Obstacles to artificial life
The main obstacle is that we do not have a good enough mechanistic understanding of the processes in natural life. The main biochemical molecules are known, the metabolic processes are worked out, but the interesting control apparatus is not understood in even the most basic way.
The reason is that the actual information in the molecules is not the chemical structure, but in their dynamical conformations, and bindings. These conformations and bindings carry data about the state of the cell, and the number of state variables in a single cell carried by the proteins alone potentially rivals a small 1980s microcomputer. This is much larger information capacity than that in ordinary dynamical systems, and with good reason. Biological systems are themselves information devices, they are computers, and the dynamics is only properly describable in the proper level of abstraction, which matches the level at which the cell itself carries and manipulates the information.
The protein conformations and bindings can be enumerated and mapped out in a given organism, but to artifically design new proteins, you have to have a protein-domain library each part of which has a known function in a given protein context. Protein domains are parts of proteins you can mix and match in order to provide new function. We do not have a protein domain library.
RNA computing
The second obstacle is that the majority of the computation in eukaryotic cells is certainly done by RNA, and the mechanism of the computation is unknown. This is not believed by the majority of biologists, but there are more who do now than before. The RNA world is not understood even at the most basic level. But I am sure that it is the products of the RNA world in eukaryotic cells that are most significant for controlling the cell.
All of this is essentially biology, not physics, so I will stop here.
A: I study nanobiology (beginning second year Bachelor) at the TU Delft in collaboration with Erasmus University (Netherlands). I would highly suggest this study if nanorobotics is interesting to you. A cell is just a collection of highly developed nanomachinery working in harmony to keep the vessel running, in the eyes of a true nanobiologist, that is. 
To just give you an example, antibodies are the (mindless) robots that enable the immune system to fight cancer (when it has gotten out of hand). You could identify membrane proteins which are markers for that type of cancer, use a computer program to literally design an antibody that attaches to this membrane protein of interest and then let the immune system do the rest. 
Obviously, this is a gross oversimplification, but the point still stands. You could, in theory, do this. 
So yes, personally, I would highly recommend you to see cell biology as natural nanotechnology and pursuit a study in the field of cell physics (preferably nanobiology). 
A: The first thing one needs to put on the table to discuss the viability of making nanorobots is a model for how the nanorobots will be created! Only then can we discuss the obstacles in principle or practice (lest we end up arguing nonsense because we're imagining totally different models).
Ron makes an excellent point about repurposing existing biological "machines," and I think this is one of the more workable models, though Ron outlines the obstacles there.
If you'd rather imagine that because we know everything about chemistry, we could just engineer some sort of assembler that puts everything where it needs to be, I suggest you read this article, which is a series of arguments between two experts:
http://pubs.acs.org/cen/coverstory/8148/8148counterpoint.html
(And from that article, with respect to biology-inspired nano-engineering, I think this is a particularly useful quotation that aligns with Bobz point about materials choice):

"And what kind of chemistry can it do? Enzymes and ribosomes can only
  work in water, and therefore cannot build anything that is chemically
  unstable in water. Biology is wonderous in the vast diversity of what
  it can build, but it can't make a crystal of silicon, or steel, or
  copper, or aluminum, or titanium, or virtually any of the key
  materials on which modern technology is built. Without such materials,
  how is this self-replicating nanobot ever going to make a radio, or a
  laser, or an ultrafast memory, or virtually any other key component of
  modern technological society that isn't made of rock, wood, flesh, and
  bone?"

