Test if a port on a remote system is reachable (without telnet)

Nice and verbose! From the man pages.
Single port:

nc -zv 127.0.0.1 80

Multiple ports:

nc -zv 127.0.0.1 22 80 8080

Range of ports:

nc -zv 127.0.0.1 20-30

Bash has been able to access TCP and UDP ports for a while. From the man page:

/dev/tcp/host/port
    If host is a valid hostname or Internet address, and port is an integer port number
    or service name, bash attempts to open a TCP connection to the corresponding socket.
/dev/udp/host/port
    If host is a valid hostname or Internet address, and port is an integer port number
    or service name, bash attempts to open a UDP connection to the corresponding socket.

So you could use something like this:

xenon-lornix:~> cat < /dev/tcp/127.0.0.1/22
SSH-2.0-OpenSSH_6.2p2 Debian-6
^C pressed here

Taa Daa!

Netcat is a useful tool:

nc 127.0.0.1 123 &> /dev/null; echo $?

Will output 0 if port 123 is open, and 1 if it's closed.