Here is a list of some of the culprits behind kernel32 errors and, with each one, a suggestion that might cure the PC of that particular affliction.

Your password file is corrupted.
If you are getting the message while trying to access your Internet service provider, the source of your kernel32 woes may be a corrupted password file.

Open Windows' Find utility and put the following, without the quotes, in the Named box: "*.pwl" and click on Find Now.
When Find has spotted the file, right-click on it and, from the pop-up menu, choose Delete.
When you restart your PC, a new password file will be created.
The downside of this cure is that, the first time you undertake functions that are password protected, you will have to re-enter each secret code.

A program and Windows do not get along.
If you get the kernel32 error only when accessing one specific program, it may be because Windows and the program are allergic to each other.

Check with the program's vendor. A patch may be available.

You are running two similar programs at the same time - two antivirus programs, for instance. Or both Microsoft Outlook and Outlook Express.

Don't.

You have a corrupted or out-of-date video driver.
Reinstall the driver, although a better idea might be to check the Web site of your video card's manufacturer and download an updated driver.

One of your memory modules has gone bad.
This is a long, long shot, but worth checking out.
If your computer does not automatically do a memory check on bootup, go to Start/Run. In the Open box, type "sysedit" without the quotes, then click on OK.

A series of cascading windows, each one containing a file Windows uses to set up and control your PC, will open. Click on the window with config.sys in the title bar.

Place your cursor in the text area and type in the following, remembering to leave a space between the sys and the slash mark:

device=c:\windows\himem.sys /testmem:on
Save and close the file. Exit from System Editor.
Adding that line will force the computer to check the memory modules when you boot up.
Don't expect the check to turn up a faulty memory chip the first time out of the gate. It may take several checks to find the culprit, if there is one.

The memory test adds several seconds to the bootup process, so don't forget to remove the line from config.sys after you have finished your testing.

Windows may be having trouble pushing your work through the video card at optimal speed.
Go to Start, Settings and Control Panel.
In the Control Panel, click on the System icon and when the Properties dialog box opens, click on the Performance tab.
Click on the Graphics button.
In the Advanced Graphics Settings dialog box, move the slider back two notches. Click on OK and restart your computer.

You may be running too many programs at the same time.
Minimize the number of programs you have working at the same time, especially if they are the sort that eat up a lot of memory: Photo-editing and home-publishing programs are two likely suspects.

If the kernel errors go away, check the status of your hard drive. If it is more than 90 percent full and/or badly fragmented, free up space until at least 10 percent of the drive is free, then defragment.

That will give Windows more room to create its swap file, which it uses as temporary memory.
You should also consider adding RAM - random access memory modules.

You have Internet Explorer 5.1 problems.
If Internet Explorer is installed incorrectly or if one of IE's shortcuts has been deleted or does not work, kernel32.dll messages start popping up all over the place.

You will have to repair Internet Explorer. To get a blow-by-blow description of this somewhat involved process, go to http://support.microsoft.com/support/kb/articles/Q194/1/77.ASP?LN=EN-US&SD=gn&FR=0