Blowing the shofar on Rosh Hashanah (the Jewish New Year) is a mitzvah (a commandment.) Women are not obligated in time-bound positive mitzvot, because of their overriding family responsibilities. However, it is certainly meritorious for a woman to hear the sounds of the shofar on Rosh Hashanah. A woman who hears the shofar being blown every year on Rosh Hashanah is considered to have taken a vow to do so, and should continue hearing it every year. There are some authorities who teach that Jewish women as a whole have accepted this commandment upon themselves.
Can a woman blow the shofar? She can certainly blow it for herself. Since her level of obligation in the mitzvah is different than that of a man, she cannot blow for a man. There are various rabbinical opinions on whether a woman can blow for other women.
This is all concerning the blowing of the shofar on Rosh Hashanah. A woman can definitely blow the shofar on other occasions during the year.
Mrs. Rochel Chein