What are the next generation physics experiments? The LHC and LIGO are two recent examples of hugely ambitious experiments in fundamental physics, both of which took decades to develop.
What are the next major experiments currently being planned and developed? What will they measure? What impact are they expected to have? And when are they expected to go live?
One example:


*

*eLISA due 2034
Developed by the ESA, eLISA will be the first dedicated space-based gravitational wave detector. Consisting of three probes spanning millions of kilometres, it will provide a hugely more accurate window to gravitational waves.
Possible signal sources: the usual GW stuff, the early phase of the big bang, and even speculative objects like cosmic strings.
I suggest, at some point, we collate all of the answers into a single community post.
 A: Facility for Antiproton and Ion Research

The Facility for Antiproton and Ion Research (FAIR) is an international accelerator facility under construction which will use antiprotons and ions to perform research in the fields of: nuclear, hadron and particle physics, atomic and anti-matter physics, high density plasma physics, and applications in condensed matter physics, biology and the bio-medical sciences. It is situated in Darmstadt in Germany and is expected to provide beams to the experiments from 2018 onwards.

A: European x-ray free electron laser

The European X-Ray Free-Electron Laser (European XFEL) is an X-ray research laser facility currently under construction and as of 2015 is scheduled to start user operation in 2017. The international project with 11 participating countries (Denmark, France, Germany, Hungary, Italy, Poland, Russia, Slovakia, Spain, Sweden and Switzerland) is located in the German federal states of Hamburg and Schleswig-Holstein. A free-electron laser generates high-intensity electromagnetic radiation by accelerating electrons to relativistic speeds and directing them through special magnetic structures. The European XFEL is constructed such that the electrons produce x-ray light in synchronisation, resulting in high-intensity x-ray pulses with the properties of laser light and at intensities much brighter than those produced by conventional synchrotron light sources.

A: It's not as fundamental as the other mentioned projects, but I am excited about ITER. If everything goes according to schedule, ITER should be the first fusion reactor that produce more energy than it consumes in 2030.
A: A large portion of the 'big science under developement' is directed towards astrophysics and cosmology. The Square Kilometre Array (SKA) and the European Extremely Large Telescope (E-ELT) are the two flagship facilities for ground-based astronomy in the future. Both are planned to be operational in the twenties of this century.
 
SKA - artist impression (source: Wikipedia). With a collecting area of approximately 1 km^2 it will dwarf all other radio telescopes.
 
E-ELT - artist impression with VLT and Colloseum added for scale (source: Wikipedia)
A: I'm really excited about the results of Fermilab and J-PARC on the measurement of $(g-2)_\mu$, that is, the anomalous magnetic moment of the muon. The current value of $g-2$ is
\begin{align}
a_\mu^\mathrm{exp}&=0.001\;165\;920\;91(63)\\
a_\mu^\mathrm{SM}&=0.001\;165\;917\;64(52)
\end{align}
where $\mathrm{SM}$ is the full Standard Model prediction, and the uncertainty $(52)$ is mostly just hadronic. There is a $4\sigma$ deviation between theory and experiment, which leads to three possible explanations:


*

*The experimental result is wrong: the errors are underestimated or there are undetected systematic errors in the measurement.

*The theoretical calculation is wrong: there is a lot of research about the hadronic contribution because it is very difficult to estimate from first principles. There is a (IMHO, high) chance that the hadronic contribution is miscalculated.

*Beyond Standard Model physics: there are unknown particles that contribute to $a_\mu$ (e.g., supersymmetric particles).
There are many planed experiments to constraint the second possibility$^1$, and Fermilab and J-PARC intend to rule out the first one, so that we are certain that the third case is the right one. Therefore, after Fermilab and J-PARC we will probably have the first quantitative evidence of BSM physics!
Fermilab is supposed to be running from 2017 to 2018 and will present the results soon after. AFAIK, there is no announced date for J-PARC, but it is expected to begin in the late 2010s.
For more info, see http://arxiv.org/abs/1512.00928
$$ $$ $$ $$
The Fermilab muon ring:


$^1$ and I hope there'll soon be definitive lattice calculations that will settle the uncertainty.
A: The Gaia Spacecraft is another hugely anticipated physics experiment. First conceived of in the early 90's it has been operational since 2013. 

The aim of this ambitious experiment is to create a 3D map of the location and velocity of up to 1% of all objects in the Milky Way. This should enable us to refine our models on galactic dynamics and allow us to investigate problems therein, i.e. pesky Dark Matter. 
Other objectives listed on the wiki page include inferring the structure of spacetime through the detection of bending photon paths and identifying/classifying astronomical objects including quasars.
We saw some initial data last year and much more is expected later this year! Below is the star density map released in 2015:

A: There are plans for a linear collider of electron positron, to study the new physics that is appearing at the LHC, two are in design.

The International Linear Collider (ILC) is a proposed linear particle accelerator.1 It is planned to have a collision energy of 500 GeV initially, with the possibility for a later upgrade to 1000 GeV (1 TeV). The host country for the accelerator has not yet been chosen and proposed locations are Japan, Europe (CERN) and the USA (Fermilab). Japan is considered the most likely candidate, as the Japanese government is willing to contribute half of the costs, according to the coordinator of study for detectors at the ILC . As of June 2013 Construction could begin in 2015 or 2016 and will not be completed before 2026.
Studies for an alternative project called the Compact Linear Collider (CLIC) are also underway, which would operate at higher energies (up to 3 TeV) in a machine with comparable length as the ILC.

A: European Extremely Large Telescope

The European Extremely Large Telescope (E-ELT) is an astronomical observatory and the world's largest optical/near-infrared extremely large telescope now under construction. Part of the European Southern Observatory (ESO), it is located on top of Cerro Armazones in the Atacama Desert of northern Chile. The design comprises a reflecting telescope with a 39.3-metre-diameter segmented primary mirror and a 4.2-metre-diameter secondary mirror, and will be supported by adaptive optics, six laser guide star units and multiple large science instruments.[8] The observatory aims to gather 13 times more light than the largest optical telescopes existing today, be able to correct for atmospheric distortions and provide images 16 times sharper than those from the Hubble Space Telescope.[9]

A: James Webb Space Telescope

The James Webb Space Telescope (JWST), previously known as Next Generation Space Telescope (NGST), is a flagship-class space observatory under construction and scheduled to launch in October 2018. The JWST will offer unprecedented resolution and sensitivity from long-wavelength (orange-red) visible light, through near-infrared to the mid-infrared (0.6 to 27 micrometers), and is a successor instrument to the Hubble Space Telescope and the Spitzer Space Telescope. While Hubble has a 2.4-meter (7.9 ft) mirror, the JWST features a larger and segmented 6.5-meter (21 ft) diameter primary mirror and will be located near the Earth–Sun L2 point. A large sunshield will keep its mirror and four science instruments below 50 K (−220 °C; −370 °F).

And when we are talking about space programs, any successor of the ISS will probably be the most expensive scientific construction project of the next decade.
