This a simple tutorial one can follow to define or reproject (transform) a Coordinate Reference System (CRS) in ArcGIS 10.1+ starting from a CSV file.

3.2. On Coordinate System of Input Coordinates field, select current CRS from data.

Now, let's create columns with new X and Y coordinates in the Table of Contents (TOC):

6. Go back to the TOC and right click X_coords;

This a simple tutorial one can follow to define (overwrite) or reproject (transform) a Coordinate Reference System (CRS) in ArcGIS 10.1+ starting from a CSV file. To clearly understand 'define' and 'reproject' concepts refer to this answer.

3.2. On Coordinate System of Input Coordinates field, select current CRS from data (or choose one manually).

Now, let's create columns with new X and Y coordinates in the layer's table of attributes:

6. Go back to the Attribute Table and right click X_coords;

This a simple tutorial one can follow to define/transform geographic coordinate system and/or datum in ArcGIs 9.3., starting from a ".csv" file.

1. Open ArcMap;
2. In ArcMap, click on “Tools” button located in Main Menu. Choose the following option: “Add XY Data”;
3. On “Add XY Data” window, click at the browse button to open the “.csv” file which contains geographic coordinates (columns with x coordinate values and columns with y coordinate values) you want to change/transform (from lat. long. or UTM -> to UTM or lat. long.);
4. Address X and Y fields on “Add XY Data” window with the respectively columns in the “.csv” file;
5. On “Coordinate System of Input Coordinates” field (still on “Add XY Data window), select the current datum and coordinate system which your input data are associated;
6. Next, click on “ArcToolBox” button --> go through “DataManagementTools” --> “Projections and Transformations” --> “Feature” --> “Project”;
7. In “Project” window: i) click on “Input Dataset or Feature Class” field (it should appear as an option the input file, that was opened in step 4). ii) in the field “Output Coordinate System” choose the desired (new) geographic coordinate system and datum to be associated with data;
8. Click with the mouse right button on the new layer that was created after previous step, and click “Open Attribute Table”;
9. It will open the window “Attributes of...”. Click on “options” (lower right corner) and next, click on “add field” (it is going to create columns which will be filled in with new coordinates x and y). Chose a name for the column and set “type” field as “Double”. This step requires repetition (for column with x coordinates and column with y coordinates);
10. Then, go back to “Attributes of…” window and give a right click on the new column name created at the previous step. Select “Calculate Geometry”. Verify if the option “use coordinate system of the data source” is selected. If yes, click “Ok”. If no, change to this option and then click “Ok”;

This a simple tutorial one can follow to define or reproject (transform) a Coordinate Reference System (CRS) in ArcGIS 10.1+ starting from a CSV file.

1. Open ArcMap;

2. Click on File --> Add Data --> Add XY Data;

3. Browse the CSV file which contains column A with longitude (or Easting) values; and column B with latitude (or Northing) values. Example:

       easting     northing  
    732771.529  7457694.818  
    733006.267  7456350.588  
    735506.391  7481351.041  
   

3.1. Select column A as the X field and column B as the Y field from the CSV file;

3.2. On Coordinate System of Input Coordinates field, select current CRS from data.

Here the CRS is defined. One can export the layer to a shapefile by right clicking in the layer and then, Data --> Export Data.

Or continue (from item 3.2) to reproject:

4. Click on ArcToolBox button (little red toolbox icon), then go through DataManagementTools --> Projections and Transformations --> Feature --> Project;

4.1. Select the input layer from the Input Dataset or Feature Class field;

4.2. In Output Coordinate System choose the CRS one wants data to be projected (transformed) to.

4.3. Set file path and name for the output file in Output Dataset or Feature Class field.

4.4. Click Ok and wait while Project finish running.

Here data is already reprojected to new CRS.

Now, let's create columns with new X and Y coordinates in the Table of Contents (TOC):

5. Right click the new created layer and select Open Attribute Table;

5.1. Click on Table Options (upper left corner), then Add field to add new X coordinates;

5.2. Chose a name for the column (e.g. X_coords) and set the Type field as Double.

5.3. Repeat steps (5 to 5.2) for creating column Y_coords;

6. Go back to the TOC and right click X_coords;

6.1. Select Calculate Geometry. Set Property for X_coords equal X coordinate of Point. Verify if Use coordinate system of the data source is selected. Click Ok.

6.2. Repeat steps 6 and 6.1 for column Y_coords, but select Y coordinate of Point instead.