The paper gives an introduction to theory and application of multivariate and semiparametric kernel smoothing. Multivariate nonparametric density estimation is an often used pilot tool for examining the structure of data. Regression smoothing helps in investigating the association between covariates and responses. We concentrate on kernel smoothing using local polynomial fitting which includes the Nadaraya-Watson estimator. Some theory on the asymptotic behavior and bandwidth selection is provided. In the applications of the kernel technique, we focus on the semiparametric paradigm. In more detail we describe the single index model (SIM) and the generalized partial linear model (GPLM).