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The Calendar class depends on a lot of Java classes that GWT cannot possibly convert to Javascript. If it fits your needs, you could simply do new Date() on the client side. See here for more details.
You could also try the gwt-calendar project.
java.util.Date will still work, and you can use com.google.gwt.i18n.client.DateTimeFormat to do any date formatting you need to do.
Check that link for full example code.
https://developers.google.com/web-toolkit/doc/latest/RefJreEmulation you can't use everything in GWT .... !