A bicycle rack assembly comprising: a rack having multiple lateral portions which are located adjacent to each other and facing to each other, one of the lateral portions having an engaging recess defined therein; a base having multiple outer portions which are located corresponding to the lateral portions, at least more than two of the outer portions being in contact with at least more than two of the lateral portions and friction being formed therebetween, the base having a connection portion which is engaged with the engaging recess; a receiving member removably connected to the base, and the rack including a frame and two bars transversely connected between two sides of the frame, a portion of the lateral portions being formed on the frame, and another portion of the lateral portions being formed on the two bars, the engaging recess being defined in one of the two bars, characterized by the base including a side recess which is located opposite to the connection portion, a positioning assembly being removably inserted into the side recess and including a movable member and a spring, the movable member being movable in the side recess, the spring being biased between an inside of the side recess and the movable member, the movable member including a tongue which is engaged with a snap recess of one of the bars, wherein, when the movable member moves in the side recess, the spring is compressed and the tongue is disengaged from the bar having the snap recess, wherein the base includes multiple blocks connected to corners of the base and each block has the outer portions adjacent to each other, wherein the engaging recess has a narrow area and a wide area, the narrow area of the engaging recess includes an opening, the wide area of the engaging recess includes another opening which is wider than the opening of the narrow area, the connection portion is an inverted T-shaped member and includes a first part and a second part, the first part is a tapered part, the connection portion is engaged with the engaging recess, the second part is engaged with the wide area, the first part is engaged with the narrow area. a rack having multiple lateral portions which are located adjacent to each other and facing to each other, one of the lateral portions having an engaging recess defined therein; a base having multiple outer portions which are located corresponding to the lateral portions, at least more than two of the outer portions being in contact with at least more than two of the lateral portions and friction being formed therebetween, the base having a connection portion which is engaged with the engaging recess; a receiving member removably connected to the base, and the rack including a frame and two bars transversely connected between two sides of the frame, a portion of the lateral portions being formed on the frame, and another portion of the lateral portions being formed on the two bars, the engaging recess being defined in one of the two bars, characterized by the base including a side recess which is located opposite to the connection portion, a positioning assembly being removably inserted into the side recess and including a movable member and a spring, the movable member being movable in the side recess, the spring being biased between an inside of the side recess and the movable member, the movable member including a tongue which is engaged with a snap recess of one of the bars, wherein, when the movable member moves in the side recess, the spring is compressed and the tongue is disengaged from the bar having the snap recess, wherein the base includes multiple blocks connected to corners of the base and each block has the outer portions adjacent to each other, wherein the engaging recess has a narrow area and a wide area, the narrow area of the engaging recess includes an opening, the wide area of the engaging recess includes another opening which is wider than the opening of the narrow area, the connection portion is an inverted T-shaped member and includes a first part and a second part, the first part is a tapered part, the connection portion is engaged with the engaging recess, the second part is engaged with the wide area, the first part is engaged with the narrow area.