Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
The basic goal of this build is to have Galio:The sentinel's sorrow a blast mage rather than being a mage. The mastery's selected all push towards mana, mana regen, ability power, and cooldown reduction. All of the runes that were chosen either put up magic penetration, mana, or ability power for many spells and lots of damage. The items each have a purpose. the meki pendent is the first item so that you can have some mana regen and also lower the price of the Chalice of harmony. The chalice of harmony helps galio gain more mana for alot of casting without having to go back to the nexus. Mercury treds give you speed, magic penetration, and magic resistance, which in turn will increase Galio's ability power due to his character passive. The abyssal scepter also give you magic resistance and ability power, which is just a double boost on galio. The archangles staff provides health and mana for galio. The Zhonya's ring will give an additional 120 ability power plus bonus for the ability power that you have. Lastly the lich bane generally just gives gailo bonus speed, mana, ability power, and magic resistance which translates to bonus ability power for Galio. HAVE FUN : )