Does Quantum Mechanics assume space and time are continuous? I was confused when I was listening to a Quantum Mechanics lecture online. Are space and time assumed to be continuous or discrete in Quantum Mechanics?
I can see the question is vague, but this is so since I am confused.
 A: All verified quantum mechanical theories till date assumes space and time are continuous. 
But in several approaches of quantum gravity theories (like LQG) spacetime may assume discrete structure. On the other hand in string theory spacetime is assumed continuous. 
There are controversies and animosity between different camps of quantum gravity research. All these approaches have the shortcoming till now in that they have failed to produce any exclusive verifiable predictions (means not something like S.T. predicts Equivalence principle etc.).
The fair and honest answer is no body knows for sure whether spacetime is ultimately continuous or discrete. Only time will tell.
A: No, space and time are continuous in ordinary quantum mechanics (not QFT or quantum gravity or string theory or whatever). Keep in mind, though, that this does not constitute a prediction that spacetime is actually continuous. Basic quantum mechanics has nothing to say on that matter. It's simply that the theory as it is normally taught and used is constructed on a continuous manifold.
A: In quantum mechanics, space may be continuous, discrete, or unmodelled. For example, in quantum information theory, one generally works in finite-dimensional Hilbert spaces, where it is impossible to define momentum and hence space translation. And in the statistical mechanics of solids, space is treated as a discrete lattice representing the crystal sites.
Time is usually assumed to be continuous, namely whenever one works with a Schroedinger equation. However, there is no theorem that QM is incompatible with discrete time. Indeed, one can easily discretize standard quantum mechanics by measuring time in units of a minimal time $t_0$ and replacing the continuous Schroedinger dynamics with the discrete dynamics $\psi_{t+1}= U \psi_t$, where $U$ is a unitary operator. The predictions are identical with the usual formalism if one takes $U=e^{-it_0H}$. 
The standard model (and hence the current agreed most fundamental experimentally verified level of description) assumes that both space and time are continuous.
In quantum gravity, people play with different formulations, some of which (e.g., loop gravity) assume a discrete space-time, whereas others (e.g. string theory) don't. None of them gives experimentally verified predictions beyond those incorporated in either the standard model and in classical gravity, hence it is far too early to decide which of the underlying assumptions are morst useful. Thus different physicists will give different anwers based on personal preferences.
A: Space and time are continuous, in quantum mechanics or otherwise. In particular, whenever our theories of any kind talk about time, it is always a real continuous parameter.
Similarly, spatial positions of particles in ordinary quantum mechanics are operators $\hat x$ whose eigenvalues are continuous, too. This fact is related to the continuity of time by the Lorentz symmetry required by the special theory of relativity.
The word "quantum" in "quantum mechanics" is often misunderstood. It refers to the fact that some quantities - such as the angular momentum and energy of bound states (pairs of particles orbiting each other, for example) - may be quantized i.e. take discrete values. However, it's not true that all quantities have to be discrete. Positions and momenta of particles are continuous. 
(Momenta may become discrete, much like energy, but only if particles are propagating in a finite region of space.)
Many people have invested thousands of manhours to the research of the possibility that space or even time are discrete. However, the results of their research have only reinforced the conclusion that such theories are inevitably inconsistent.
A: There is fundamental problem with the meaning of the notion of "discrete time and space for QM". In QM if You need to look at some very small structure, You have to use very large energy. So answer to Your question may have unintended effects included. I think about situation when someone asks about how it looks like space and time form very small time intervals and small spatial structures but do not think that it implies very large energies at all! 
If someone do not think about it it may lead to false conclusions, that "normal space and time" may be discrete all the time, whilst this may be not the case. At normal energies we probably may safely say that space and time are continuous enough to do not worry about any discretization because of various symmetries which are used in quantum mechanics during description of interaction with matter, about which Lubos mentioned above. 
From the other hand if discretization of space and time will be required in order to describe effects of such interaction for very large energies - it may be not satisfactory answer to Your question, because it may be an effect ( for example unification of QGravity and QMechanics) which may appear only during early stages of Big_Bang and then do not lead to any results which may be valid now -  where gravity and quantum world are rather weekly coupled. 
Then it may important philosophical and historical meaning but may not have any any practical meaning in a sense that there may be no way to see any effects of this discretization here and now - then QM may not need any "correction from discrete time and space". 
A: 
...(the idea) "that space is continuous is, I believe, wrong."

— Professor Richard Feynman 
The Messenger Series: Seeking New Laws
A discontinuous space | continuous meta-space model fits observed facts, and provides refreshing insights into many aspects of life.
A: Sure it's possible to formulate QM with discrete time. For instance, take a quantum circuit constructed from reversible quantum gates.
