Curvature of Spacetime I have been exploring  for some time both the Special and General Relativity, hoping to glean at least a conceptual grasp of their basic tenets.
In reading the book "Gravitation" by Misner, Thorne and Wheeler, the authors stress that Riemann came very close to make a decisive connection between gravitation and curvature of space, but he failed to do so, they say,  precisely because he thought of SPACE and curvature of SPACE  instead of curvature of SPACETIME and this makes the whole difference! 
Can anybody explain in conceptual terms, as far as possible, why spacetime, unlike SPACE alone, can be seen and understood as curved? 
I firmly believe that mathematics is only a language, albeit a complex one,  which facilitates our understanding of reality, but that the same reality is not hopelessly beyond reach  without maths. We should remember after all that Einstein's mathematical formulations of the Special and General Relativity are rooted in thought experiments and in a basic conceptual grasp, which preceded its mathematical formulation.
 A: Gravity must be understood as a curvature of spacetime rather than space itself because the 1915 general theory of relativity, Einstein's new theory of gravity, is an extension of the 1905 special theory of relativity and the special theory of relativity introduces an inseparable connection between the space and the time and forces us to talk about them in a unified – talk about spacetime.
Space and time have to mix according to special relativity because the theory starts from two postulates, including the absolute constancy of the speed of light in the vacuum, and if space and time were separated, such a constancy would be incompatible with the other postulate, the identical form of the  physical laws as seen by an arbitrary inertial observer. It makes no sense to discuss a better, post-Newtonian theory of gravity without taking special relativity into account; the general theory of relativity with its insights about the spacetime curvature is a result of the reconciliation of Newton's gravity and special relativity.
In fact, when one studies how Newton's approximate (inverse square) laws of gravity emerge from general relativity, it turns out that the "curvature of time", and not so much "curvature of space", as a function of space plays the decisive role in determining the gravitational fields at each point. Technically speaking, the rate of time at a given point is determined by $g_{00}$ which is approximately a linear function of the gravitational potential $\Phi$ known from Newton's theory.
Conceptual ideas may precede the mathematical formulation of some principles but one usually can't get too far if he avoids mathematics. Well over 99% of important insights in modern physics depend on mathematical equations and structures that may be at most translated to "awkward and confusing" words.
A: Space can be curved alone, but that does not correspond to our universe.  The surface of a sphere is a nicely curved manifold without boundary.  One can imagine a universe where that was space (or maybe a 3-sphere) with the absolute time of Newton.
A: The reason why space-distortion alone cannot describe gravity, is that space-distortion can only affect objects which already move through space. But gravity also affects objects which are initially at rest in space. Objects at rest in space still advance in time, so you have to combine space & time into one structure, like shown here:
http://www.youtube.com/watch?v=DdC0QN6f3G4 
The distortion on the time axis describes the gravitational time dilation. Clocks placed lower in the gravitational field, run slower:
http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/principal_of_equivalence_and_general_relativity/curved_spacetime.html
A: General Relativity inherits the space+time structure of Special Relativity, where space and time are linked together and represent different directions along which one can move at the same speed, the speed of light. For a reference frame at rest with you, you move only along time, but for a moving reference frame you convert part of your speed towards spatial directions, decreasing the speed you move in time. This is the time dilation effect.
Now, in Einstein's description, gravity is not a force but a property of space-time. Mass and energy change the structure of space AND of time, distorting both, and creating what is called a "curved" space-time: in Newtonian terms, gravitational fields dilate (slow down) time and shrink space. It turns out that the space shrinking effect is much smaller than the time dilation one, and is only relevant for things moving at relativistic speed, like photons. The slowing-down of time is much larger and so, as Lubos very correctly pointed out, is the main effect of Newtonian gravity.
You can picture it quite clearly (at least to my opinion) if you imagine a planet (or test body) moving. In empty space, the body will follow a straight trajectory say with speed v. If you now place a massive star say on the right hand side of the moving body, then time will flow slower on the right of the body than on the left:
dt_right < dt_left
So, during an infinitesimal time interval dt, the right and left sides of the planet will advance respectively by
ds_right = v * dt_right < v * dt_left = ds_left
and the body will turn right, similar (but different in the details) to what happens when
you make a turn driving a car - and in fact the car differential takes care of that differential rotation.
Hope that this helps!
A: The curvature of spacetime is defined by distortions of distance and time measurements. The curvature does not bend spacetime in a 5th dimension. 
This is often illustrated by considering the surface of the Earth as an example of a curved space. It is true that the 2 dimensional surface of the Earth does bend in the 3rd dimension. So in this regard, the example is misleading. But in other ways, it is a good example. 
Suppose the surface of the Earth was flat, like a paper map. You could go from one corner of a square to the opposite corner two ways, north then east, or east then north, and wind up at the same point. 
That doesn't work on the Earth. It is easier to see if you use a big square. Start on the equator. Go north to the pole. Turn left and go south to the equator. Try it the other way. Go 1/4 of the way around the world to the east. Turn right and go to the pole. The two routes lead to two widely different points. 
Arriving at two different points is the hallmark of curvature. 

GR says that spacetime is curved near matter (Actually near energy density, pressure, and shear stress.) One effect of this is that time runs slower in a gravity well. 
A "square" in spacetime can have one distance side and one time side. Suppose you are above a neutron star. The other corner of the square is the place and time straight below you on the surface, one second from now. 
There are two routes. You can find the point on the surface below you right now, and then wait one (slowed) second. Or you can wait a second, and then find the point that is right below you at that time. The two routes lead to different place-and-times.  
There are also purely spatial distortions. That is, there are distortions in spacetime where all sides of the square are distances. If you wanted to orbit the neutron star, you fight first want to measure the circumference of the orbit. You might also calculate the distance to the center of the neutron star from $2\pi r$. If you measure the distance to the center, you would find it is farther than $2\pi r$. 

Reimann developed the mathematics of curved space. But he only considered space where all dimensions are distance. 
A prime difference between classical physics and relativity is that time is the same for everybody in classical physics. In relativity, time has some properties in common with space. 
I am at rest. My position at time $t_0$ and $t_1$ is the same, $x_0$. You see me as moving. You say my positions at $t_0$ and $t_1$ are different. This confuses nobody. It is a simple effect of motion. 
Time has this same property. I am at rest. I am holding two lights at $x_0$ and $x_1$. The lights flash at the same time, $t_0$. 
You are moving. You say the lights flashed at different times. This too is a simple effect of motion. It ifs very confusing, but true.
Einstein saw the deep connection. Time behaves so much like space that it must be included in the distance vector. Leaving it out gives wrong answers. It would be like measuring the distance from the base of a mountain to the top on a map. You get the wrong answer if you leave out altitude. This led to SR. 
To develop GR, Einstein applied Reimann's mathematics of curved space to spacetime. 
A: When we want to study any system we are biased if we study it from inside of it.  An example: the Coriolis effect that can only be understood if we move out to a referential 'above' the rotating observer at Earth.
The Einstein relativity viewpoint is from the referential attached to an observer in motion irt to others that send/receive radiation to/from it - light, and to the background. 
When we stop the motion the 'spacetime' became space+time again. 
A viewpoint of an observer in motion is only a viewpoint, and not the exterior reality.
The observer is only 'a proper observer' that can not modify the outside of it.
An observer senses the environment with its atoms that behave like, and are, antenae. If it is in motion it will pursue its own emmited radiation, compressed wavelengts in front of it and larger wavelength in the rear (see the doppler image).
The wavelength is its unit of length and this is variable depending on the direction of reception/emission, irt the direction of its motion, and this is why the MM experiment gave a null result.  (put 2 of the above linked images of doppler travelling in parallel with the same velocity and the line connecting the mobiles 
shows a compression zone followed by a 'decompression' one, or vice-versa)  
Coincidences:
FlyBy anomaly
google '3.099e-6 anderson',read the numeric model of the FlyBy anomaly, by coincidence is an anomaly that after all the relativistic approaches are done  persists to be an anomaly. Now leave the so called inertial observer at Earth and acknowledge that the Earth is in motion at $\pm 370 km/s$ through the background (search CMB dipole) and by coincidence $$(V/300000)^2*2=3.099e-6$$ gives $V=373.4 km/s$ (% difference < 1%).  
Mercury perihelium anomaly (strong GR test)
google '43 arcseconds mathpages':  

Combined with the precession of our equinox reference line, this
  accounts for 5557 arc seconds per century, which is close to the
  observed value of 5600, but still short by 43 arc seconds per century.

another coincidence :
$$43/5557=2*pi*V/300000$$ gives $V=369.46 km/s$ that is the absolute speed of the Earth through space (a big match > 99.85%).
The trajectory of the Earth through 3D space is not an ellipse, as modeled in the textbooks, but an HELIX and this was not known by Einstein.
(the 2*pi is justified by: 'A circular helix of radius a and pitch 2πb')
And, also by coincidence, the Pioneers anomaly have the trace of $V$, allow me to keep it for myself.
This is the first time that I make public those coincidences, and by now it is not difficult to model correctly those coincidences by adopting one referential 'above' the observer, standing still in the background, what I call the referential of the Light, or the CMB referential and making a more precise definition of electromagnetic/gravitic distance. The one in use derived from half travel time of light is only an approximation.  
I'm not against Einstein, but I know that the knowledge of physics will progress as usual.  
A: there is NO curvature of spacetime. To prove this claim we will first assume that there IS curvature of spacetime. From that we derive the following: spacetime must have properties. The truth of the matter is that spacetime does not have properties. Therefore, our initial assumption was invalid, i.e. opposite must be true: there is no curvature of spacetime.
2nd proof:
if spacetime can be "curved" then, based on the law of physics which states that every action is met with reaction of the same magnitude but opposite direction, spacetime would "curve back" with equal force, i.e. there would be no curvature.
the truth of the matter is that movement of bodies in universe cannot be explained by curvature of spacetime at all. the theory should be abandoned immediately.
what about gravitation then?
there is NO gravitation. what we perceive as gravitation is essentially electromagnetic force. Earth, for example, is a moving charge. The same is true for all other planets, stars and bodies. These moving charges exert electromagnetic force on surrounding ether (yes, we have to go back to ether theory), which exerts electromagnetic force on other bodies in the universe.
