The idea that there exists planetary systems other than that of the Sun is old, but it has until recently been beyond the technical feasibility to get any observational evidence for external planetary systems. By the IRAS discovery of far-IR dust emission around nearby stars, like Vega, it was realized that the dust replenishment required larger colliding or evaporating bodies, i.e. processes similar to those that provide the zodiacal dust.
In addition, a steadily increasing number of planets are being found by detecting the minimal cyclic radial velocity variations of the central star caused by an orbiting planet. Obviously this (indirect) detection method works bests for heavy planets, orbiting close to their central stars, and there is an ongoing discussion what technical means it may take to do direct detection of planetary systems similar to our own. It has been argued that a new generation of extremely large ground-based telescopes (ELTs) may provide the tool for such observations (actually this is one of the main scientific drivers for the large investments required to build ELTs).
But probably we have to wait for space interferometers, like Darwin, for the first detection of an Earth like planet orbiting an nearby star.