Since the first attempts to put something manmade in space, we have been searching for ways to do so more efficiently. Our current system based on rocket boosters and copious amounts of multiple kinds of fuel are not only inefficient but also prone to disaster, as we have seen a number of times. Moreover, our reentry strategies are rather crude to boot; overall, our mechanism of entry and exit from space could undoubtedly use an overhaul.
Space elevators could be the innovation that delivers a novel way of putting human or machine into space. The concept dates back a number of years, but only became scientifically feasible in the 1990s, in part due to the discovery and fabrication of carbon nanotubes. 
A number of designs have been suggested. One is to simply build a rigid structure from the ground all the way up through the upper atmosphere and into space; the elevator car would then ride on this structure. Another, more mechanically feasible (due to material strength-to-weight ratios) idea is to attach a large counterweight to the end of a cable that reaches out into space; centrifugal force would keep the cable in tension and the elevator car would ride up the cable. While other designs have been put forth, these are the most likely candidates at this point; in this report I focus on the latter design because of strength to weight ratio issues with the former.
The most likely material for use in a space elevator is the carbon nanotube. Discovered within the last couple decades, carbon nanotubes are single- or multi-walled cylinders where the the walls are simple a graphite sheet one atom thick (also known as graphene). Because of their composition and atomic arrangement, they would have an ultimate tensile stress of anywhere between 49 and 150 GPa, compared to 5 GPa for steel, and a Young's modulus of around 1 TPa. [1-3] These numbers represent individual single-walled nanotubes; space elevators would in reality need to be built of "bundles" or some sort of arrangement of the nanotubes, which would result in these numbers being lower due to structural factors like inter-nanotube defects. 
Furthermore, carbon nanotubes are much stiffer than steel and other possible materials like Kevlar. This is pertinent because on the length scales we're considering, tapering the cable or tower is inevitable. However, with carbon nanotubes, the tapering is 8 orders of magnitude lower than Kevlar and 33 orders of magnitude lower than steel. 
One key engineering consideration is manufacturability. What is the best way of manufacturing a cable or tower long enough to reach into space as well as being robust enough to last in these extreme environments? Do we currently possess technology that will allow us to?
Of course one of the most important considerations is the mechanism of raising or lowering the elevator on the structure or cable. Will there be another cable that pulls on the car, like in common elevators? Or will there be a gear system that will allow the car to ride along the structure? These are important points to make, and at this point are still simply conceptual designs rather than empirical ones.
How would an artificial structure fair in the broad range of environments included in the expanse of a space elevator? In space, the elevator would witness harsh radiation, meteors, and other space debris; in the upper atmosphere, extreme temperatures and chemical factors (like sulfuric acid droplets); and in the lower atmosphere regular corrosion, lightning, rain, air traffic, and a plethora of other everyday factors. 
Though independent of technological constraints, these issues contribute to the engineering difficulty of the project. Choosing different materials for different parts of the elevator due to differing environmental conditions is a feat in itself. Perhaps as carbon nanotube technology continues to develop, it will become apparent that none of these problems need be considered; for now, all probable effects must be accounted for.
Our next trip to space will definitely not be on a space elevator. There are a considerable number of obstacles remaining to implementing a feasible space elevator: cost, manufacturability, maturation of carbon nanotube technology, and of course politics. A great amount of work is focused on finding ways to mass produce long carbon nanotubes, and right now the longest ones are on the order of 10-20 cm.  This is pertinent because the space elevator structure would need to be around 35,000 km long in order to maintain geosynchronous orbit and to have the gravitational and centrifugal forces cancel. 
Nevertheless, the idea of space elevators is a promising one that we cannot reject simply because the concept is ahead of the technology. The coming years will be tumultuous for space travel, partly due to technology and partly due to politics. With the increase in private aerospace development however, space elevators will still continue to be developed and may truly provide a ladder to the sky.
 P. K. Aravind, "The Physics of the Space Elevator," Am. J. Phys. 75, 125 2007.
 H. W. Zhu et al., "Direct Synthesis of Long Single-Walled Carbon Nanotube Strands," Science 296, 884 (2002).
 D. Srivastava et al., "Nanomechanics of Carbon Nanotubes and Composites," Appl. Mech. Rev. 56, 215 (2003).
 N. Pugno, "On the Strength of the Carbon Nanotube-Based Space Elevator Cable: From Nanomechanics to Megamechanics," J. Phys. C 18, S171 (2006).