There were a number of problems in the late 19th Century in the United States that still exist today.  There were racial tensions during this period that were very defined.  African-Americans suffered through black codes, Jim Crow Laws, and even lynchings.  There was very little economic or political opportunity for African-Americans during this time.  While African-Americans fought for civil rights for the better part of the Twentieth Century, a disconnect between the races still exists today.  Black Americans suffer from housing discrimination, unequal treatment before the law, and are behind in terms of educational and employment opportunities.  

A number of other problems that existed during the Gilded and Progressive Eras are still issues today.  Urban decay, environmental protection, corporate greed and abuses, and the rights of women come to mind as obvious examples.