1. download latest Tomcat from http://tomcat.apache.org
2. unzip it
3. rename the directory to "Tomcat" (this just makes it nicer to handle)
4. move it to the /Library directory
5. start Tomcat with /Library/Tomcat/bin/startup.sh (the shutdown script is called shutdown.sh)
6. browse to http://localhost:8080 to see if its running

I will from now on refer to /Library/Tomcat as my $TOMCAT_HOME. By the way, I read (but can't confirm) that it helps in this process to have the system variable $JAVA_HOME set to the bin-directory containing your java/javac executables.

Axis

1. download axis-bin-X_X.zip from http://people.apache.org/dist/axis/nightly/
2. unzip it
3. copy the axis-X_X/webapps/axis directory to $TOMCAT_HOME/webapps
4. Restart Tomcat
5. browse to http://localhost:8080/axis/ to see if it works
6. browse to http://localhost:8080/axis/happyaxis.jsp to check if needed components are found. (I lacked the Java Beans Activation Framework from http://java.sun.com/products/javabeans/jaf/downloads/index.html. Unzip the archive and copy the activation.jar to $TOMCAT_HOME/lib. Restart Tomcat.)

MySQL

I will be rather short here. MySQL can easily be installed and/or there are a lot of tutorials to be found via Google. On OSX, I had previously used MacPorts to install it for me.

Juddi

1. download juddi-0.9rc4.tar.gz (or latest) from http://ws.apache.org/juddi/releases.html
2. unpack it
3. copy juddi-0.9rc4/webapp/juddi/ to $TOMCAT_HOME/webapps
4. in juddi-0.9rc4/sql/mysql/ you find scripts to set up the database (create_database.sql) and insert publishers (insert_publishers.sql). Execute the contents on your MySQL client. The latter one contains only commented template code for you to insert publishers you want (maybe only important for later).
5. get the JDBC connector library here: http://dev.mysql.com/downloads/connector/j/5.0.html, unzip it and copy the jar-file you find in it to $TOMCAT_HOME/lib
6. edit $TOMCAT_HOME/conf/server.xml, add to element "host":
   
   <Context path="/juddi" docBase="juddi" debug="5" reloadable="true"
     crossContext="true">
     <Logger className="org.apache.catalina.logger.FileLogger"
                  prefix="localhost_juddiDB_log" suffix=".txt"
                  timestamp="true"/>
     <Resource name="jdbc/juddiDB" auth="Container"
                  type="javax.sql.DataSource" maxActive="100" maxIdle="30" maxWait="10000"
                  username="root" password=""
                  driverClassName="com.mysql.jdbc.Driver"
                  url="jdbc:mysql://localhost:3306/juddi?autoReconnect=true" />
   </Context>
   Note that I used the MySQL root user here. You might want to use this in a productive environment and then, of course, you choose some other user/password combination!
7. restart Tomcat
8. browse to http://localhost:8080/juddi/happyjuddi.jsp to see if juddi is happy