Stream flow is an important factor in the study of water resource management, floods, and droughts. The characteristics of the mean monthly and mean annual flow of temporal and spatial distribution and the existence of a trend in annually, mean annual, mean monthly, maximum and MAM-7d stream flows of 22 selected rivers basin has been carried out by using nonparametric tests, Mann-Kendall and Sen’s slope estimator. This study also used to identify the probability distribution functions that are best fit to maximum, mean stream annual and low flow for each station. The most commonly used goodness of fit tests namely Kolmogorov-Smirnov, Anderson-Darling and Chi-squared are applied to the data series to check the fit of probability distributions. The highest mean monthly flow was recorded in August at 90.9% of the stations and in September at 9.1% for the rest of the stations. The long term mean annual discharge varies approximately between 0.9m3/s at Amen and 54m3/s at Gilgel Abbay stream flow stations. Spatial distribution of flows over the basin was mapped and shows the highest mean annual flow of 38.45 m3/s Northern part of the basin and the lowest mean annual flow of 7.63 m3/s in the Southeast. The Mean annual streamflow showed a positive significant trend at seven stations and a negative significant trend at fifteen stations and also nine of the twenty-two stations showed slight positive trends and twelve stations shown negative trends but the rest one didn’t shown any change in annual time series.Only two stations (Dondor and Hoha) showed increasing trends for all twelve months. Generalized Pareto, Log-normal, Logistic Log-logistics(3p), Generalized Extreme Value, Lognormal (3P) and(2P), Normal, Gumbel Min and Max, Gamma, Weibull (3P), Pearson 5 and 6, and Log-Logistic (3P) distributions fit best to the different scale of stream flow. Low flow frequency analysis was used to estimate the long-term low flow quantiles. A number of probability distributions are used to estimate low flow discharges with various extreme hydrologic events 2 to 500-year return period. Among best fit distributions were Generalized Pareto, Gumbel, Log-Logistic (3P), Generalized Extreme Value, Lognormal (3P) and Log-Pearson.