The following is a list of online resources for environmental and health data. I have visited them all and used data from several. They are mainly US, but I would appreciate any foreign sites you might find. For some of these data (e.g. NCAR) you may need to register, which is free.
I'll add to these as people suggest them.
LINKADDRESSCOMMENTSEXAMPLE
ArcExplorer www.esri.com/software/arcexplorer The WWW tab on ArcExplorer takes you to an ESRI website that adds data to your map. I'm not sure how it works, but I was able to use it to download map data for the Senegal region Senegal
CDC www.cdc.gov The US Centers for Disease Control and Prevention's A to Z Index has hundreds of links to CDC websites, many of which have data. Look for diseases/conditions of interest to you or for which there is likely to be data for states, countries.
EPA www.epa.gov/airmarkets/cmap/data/ The US Environmental Protection Agency has a GIS Download page with lots of maps, but you could find more if you start from EPA's homepage. Mean NO2
ESRI www.esri.com/data/download The Environmental Systems Research Institute has at least one site for free data, but I'm sure there are others.
EROS edcintl.cr.usgs.gov/data.html The USGS EROS Data Center is a major source of environmental datasets. I used its African Data Dissemination Server to make a cropland use intensity map of Chad. Chad
GeographyNetwork www.geographynetwork.com This is an ESRI-sponsored website. A search for 'health and disease' yielded only 3 hits, but a search on 'environmental' got 243!
Geospatial One-Stop www.geodata.gov/ This US Federal site is a new data clearinghouse that appears to be well-organized, but I haven't downloaded any of its datasets.
GRID
Arendal
www.grida.no/db Another UNDP GIS data depository. I used it to create a gridded population map of West Africa. The data may require some ArcGIS. W. Africa
population
National Atlas www.nationalatlas.gov/atlasftp.html This US site is designed to provide datasets for mapping. Some of these are compressed in .tar and .gz format, so you may need help in extracting them. Butterflies
NCAR www.gisclimatechange.org The National Center for Atmospheric Research provides shapefiles of point data for climate model runs. They can be viewed in ArcExplorer and overlayed with polygons, e.g. countries. 2011 global temperature
TopoZone www.topozone.com This site provides scanned USGS topo sheets (digital raster graphics: DRGs) and arial photographs (digital orthophotoquads: DOQQs). You have to register, but the data are free. Philadelphia
UNEP geodata.grid.unep.ch The United Nations Environment Programme has lots of environmental and health datasets. You may have to do some editing to get what you want, but this is a major site. Africa water
WHO www.who.int The World Health Organization's yearly World Health Report has an annex that can be downloaded in Excel format. I can help you link the data to a shapefile.

2005/03/31