1. Extrasolar planets - exoplanets
Planets have been discovered orbiting other stars.
The list of confirmed exoplanets has over 3000.
Evidence comes from several sources:
- Distortion of protoplanetary disks (called proplyds).
- Doppler shift of the light of the star, indicating radial velocity shifts.
- Astrometry to measure stellar wobble.
- Detection of dimming of the star, indicating a transit by a planet.

2. Beta Pictoris has a planet named b Pictoris b seen in certain images

6. Doppler effect
When a source of sound or light is moving, an observer will hear a shift in frequency.
Simulations show this:
We have a classroom demonstration of this.

7. Extra-solar Planets are revealed by the Doppler effect, which measures “radial velocity” of the star

8. Sinking Planet theory of giant planets close to a star.
Large planets may form at a large distance from the star but then they collide with the gas and dust in the protoplanetary disk, and so they lose some speed and then spiral into an orbit closer to the star.
Most of the large planets seen by Doppler measurements are in “close orbits” and are too hot to support life because they are too close to their star.
This is due to bias in the method: we see the larger planets more easily.

9. Experimental methods have their own particular bias
Experimental methods have their own particular bias. Doppler surveys find mostly the large planets. Kepler transit studies have only found short period planets.

10. Kepler has found more planets smaller than Jupiter,
so it is adding to the list of smaller planets that Doppler
measurements have not seen.

11. Searches for exoplanets.
Many searches are underway. This long list gives you some idea of the importance of this topic in Astronomy. A great deal of effort is going into these searches. link
For more about extrasolar planets, see these web sites:
The Kepler mission is the most active right now, finding many new exoplanets by looking for transits. link
There are other lists that have different criteria for a confirmed exoplanet:

12. Earth-size planets with temperatures suitable for life have been found by the Kepler spacecraft. The inset plots the planet’s radius and temperature with green dots. (from NASA Exoplanet Archive)

13. More on exoplanets
Astronomy picture of the day (APOD) pages on exoplanets:
First direct image of an exoplanet (in infrared):
Beta Pictoris b:
Fomalhaut has a planet, imaged in visible light:
Multi-planet star system (Keck infrared image):
Kepler-11 has six planets:
How common are Earth-sized planets?
A protoplanetary disk forming around a star
Super Planet Crash (a game)

14. More on exoplanets
Astronomy picture of the day (APOD) pages on exoplanets:
Gliese 832c: A Potentially Habitable Exoplanet
Earth-size Kepler-186f
The Closest Star has a Potentially Habitable Planet
An article about this
Animation of radial velocity method for finding exoplanets
ESOcast 69: Revolutionary ALMA Image Reveals Planetary Genesis
ESOcast 87: Planet found around closest Star