Usually, slows from different sources stack with each other, but multiple of the same instance do not, but refresh their duration instead. There are some exceptions to this.
Percentage Movement Speed Changes can not exceed the minimum and maximum movement speed limits and are applied after flat movement speed bonuses and reductions. There are stacking restrictions for items, but no restrictions on stacking percentage movement speed bonus from abilities. The percentage of multiple slow instances fully stack, following the movement speed formula. A unit's movement speed cannot be slowed below 100, unless the unit is affected by pathing blockers, which set affected units' movement speed to 0.
A unit affected by haste fully ignores movement speed slows.
Attack speed bonuses and reductions stack additively by a flat value. Unlike movement speed, there is currently no sources that reduces attack speed by a percentage. By default, the total attack speed value has a minimum of 20 and an maximum value of 700 and most unit's attack speeds cannot be reduced or increased beyond the values above. Disarms do not slow attack speed, but simply do not allow affected units to attack at all, so their attack speed value is unaffected.
There is currently no equivalent to haste for attack speed, although some abilities (e.g.) grants an enormously high attack speed bonus, so that it is impossible to reduce their attack speed in a regular game.