As GdD pointed out, these are just typical scanning activities that occur on a daily basis for servers that are connected to the internet. Similar to what you have observed, I have seen a recent surge in X-Mas scans from a large number of IP addresses, probably because it's near the festive season.
The first thing to do is to recognize what the attacker can find out from these scanning. The objective of TCP/UDP scanning is to map out all ports that are open on the server and determine what kind of OS your server is likely to be using. Further reconnaissance may be conducted to determine what services are running behind those open ports.
Normally, tools like Nmap or Nessus would be used to probe your server for open ports. These tools rely on your server responding to them to find out if a port is:
If your server is using an out-of-the-box firewall configuration, then more likely than not, it will be compliant to RFC 793. Scanners rely on firewalls to respond in a predictable manner to find out if a port is open or closed. If your firewall is non-compliant, i.e. network packets are dropped when they should be rejected, or vice-versa, then the scanners would get confused and probably give a wrong report.
Depending on the level of skills, your attacker may mask his IP address by deploying decoys or by scanning indirectly with zombies. Though I hope there would be fewer cowards out there who scan bravely because there is really nothing to fear helping people check if their door is locked, it is quite a futile attempt to root out the perpetrators and most people would not even bother.