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Possible security issues when running a web server on a home connection?

I want to run a home web server, but I'm concerned this could expose my home network to security issues.

The server will run on a dedicated machine, with nothing on it except the os + lamp stack.

The machine will be connected to a home modem, via ethernet, and the network will be the same of my other devices: laptop, console, smartphones etc.

I will run some websites in the server, and I'm concerned about how it's presence on the web could expose my IP address, and consequently put my privacy in danger.

What are are the security problems I could encounter exposing my home ip on the net? If my server is compromised, will my entire network be in danger?