I have got a computer with Windows 8.1 64 bit and I have installed LEGO Mindstorms EV3 software Home edition and tried to connect the EV3 brick to the computer, but the brick didn´t connect to the computer. Where is the problem?
I'm going to assume this is a Bluetooth connectivity issue because connecting the cable is trivial, and I've found connecting with Bluetooth quite tricky. This may not be the only way, however it has worked for me. The main issue is that you first need to "pair" the EV3 with your computer.
Setup (you have Bluetooth, and it is turned on):
Make sure the computer has Bluetooth turned on. On my Win8 computer, there is a Bluetooth icon on the task bar when it is active. If you don't see that icon, or don't know if Bluetooth is on, go to: PC Settings > PC and Devices > Bluetooth (also under: Network > Airplane Mode. If you can't find it, try searching for Bluetooth on the computer)
set the EV3 to have Bluetooth Visibility checked true. This is under the tools tab (far right) on the EV3. Move the cursor to the Bluetooth line, and click on that to see the Bluetooth details. Both "Visibility" and "Bluetooth" need to be checked on boxes on the right (ignore the icons on the left, they look like radio buttons but are not). Once those are checked, the EV3 will be visible to the computer's Bluetooth.
Once you've confirmed that visibility is on, and Bluetooth is working, there are a few different ways you can pair your computer with the EV3.
from the computer's Bluetooth software
under the Control Panel > Devices and Printers, you can "Add a Device"
from the Bluetooth Icon on the task bar ("Add a Bluetooth Device").
No method is better, whichever you find first. Once it starts scanning, it should display the EV3 within a minute or two. When the EV3 when is found, choose "pair" or "connect". Now, switch to the EV3 which should display:
"Connect? (your computer's name)"
Move the cursor to the check box (default is on the "x" as in "No"), then you will be shown the passkey: 1234. No need to change that. Click that check box too.
Now, switch back to your computer, where you'll be prompted for the passkey. Enter 1234, and it should complete the pairing. If you are interrupted (I believe you have 2 minutes) it'll lose the connection and you'll need to start again.
After all of that, you should be able to start the Mindstorm software, open a project, and in the lower right, choose "Available Bricks" tab, and see your EV3 (if not, click the "refresh" button in that window and it should show up). Once there is a check box for your EV3, click that box to finally connect.
The hard part is the pairing. Once paired, it is simple to connect in the future. I just make sure the EV3 is on before the Mindstorm software is started. If the EV3 is turned off, I have to click that "refresh" button telling the computer to find it again, then check the box to connect.
Final note: on the Bluetooth menu of the EV3 (where you check Visibility), there is another item "Connections." This isn't merely a title. Click on this, and it will display all computers that it has paired with in the past. You can click on these and have them removed. I've had to do this occasionally to re-establish connection with a computer. This is how you would remove old connections too, if you wanted to clean things up, or just want to make sure there aren't any left-overs from earlier connection attempts.