From the Wikipedia page  - which has links to the articles on the various systems.
Writing emerged in a variety of different cultures in the Bronze age. Examples include the cuneiform writing of the Sumerians, Egyptian hieroglyphs, Chinese logographs, and the Olmec script of Mesoamerica. The Chinese script likely developed independently of the Middle Eastern scripts, around 1600 BC. The pre-Columbian Mesoamerican writing systems (including among others Olmec and Maya scripts) are also generally believed to have had independent origins. It is thought that the first true alphabetic writing was developed around 2000 BC for Semitic workers in the Sinai by giving mostly Egyptian hieratic glyphs Semitic values. The Ge'ez writing system of Ethiopia is considered Semitic. It is likely to be of semi-independent origin, having roots in the Meroitic Sudanese ideogram system. Most other alphabets in the world today either descended from this one innovation, many via the Phoenician alphabet, or were directly inspired by its design. In the case of Italy, about 500 years passed from the early Old Italic alphabet to Plautus (750 to 250 BC), and in the case of the Germanic peoples, the corresponding time span is again similar, from the first Elder Futhark inscriptions to early texts like the Abrogans (ca. 200 to 750 CE).
See also Proto-writing