A number of missions from various space agencies are currently dedicated to understanding comets and asteroids. Below you can find a summary:
ESA's Rosetta spacecraft was launched on March 2004 to reach, study and finally land on comet Comet 67P/Churyumov-Gerasimenko. The mission consists of an orbiter and a small lander, each of which carries a large complement of scientific experiments. Rosetta flew past the main-belt asteroids (2867) Steins in September 2008 and (21) Lutetia in July 2010. Rosetta has successfully entered orbit around the comet in 2014. Its lander, Philae, was released and landed on the comet surface on October 12th 2014, after which it has carried out several investigations. The Rosetta orbiter will spend some 2 years carrying out observations of the nucleus as the comet heads towards the Sun.
To know more: http://www.esa.int/SPECIALS/Rosetta/
Dawn is a NASA mission launched in 2007. The Dawn mission will undertake a journey in both space and time by traveling to the two oldest and most massive asteroids in our solar system: Vesta, that will be reached in 2011 and Ceres, where Dawn will arrive in 2015.
The mission's science goals are to study and compare these two very different bodies, one cool and wet, the other hot and dry, to understand the conditions and processes in place at the beginning of solar system formation. Dawn's science instruments will measure the asteroids mass, shape, volume, spin state and mineral composition, will determine tectonic and thermal history, magnetism and core size and examine the internal structure of the two minor bodies.
Hayabusa-2, JAXA's second asteroid explorer, was successfully launched on December 3, 2014. The Hayabusa-1 mission captured sample particles from an asteroid and brought these back to Earth in 2010. Capitalising on the experience from the first spacecraft, Hayabusa-2 aims at acquiring samples from a C-type asteroid (1999JU3) to elucidate the origin and evolution of the solar system and of the composing elements.
To know more: http://global.jaxa.jp/projects/sat/hayabusa2/index.html